A HISTORY AND DEVELOPMENT OF IND‘ECOMPOSABLE CONTINUA THEORY Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY FRANCIS LEON JONES 1971 L IIIIIIIIIIIIIIIIIIII 05 Michigan Sm: University This is to certify that the thesis entitled A History and Development of Indecomposable Continua Theory presented by Francis Leon Jones has been accepted towards fulfillment of the requirements for Ph 0 D a degree in Mathematic S o/wQ “415/31 9‘ ham professor / Date Jul 29 19 0-7639 92%. I Iris III.I II ABSTRACT A HISTORY AND DEVELOPMENT OF INDECOMPOSABLE CONTINUA THEORY By Francis Leon Jones This thesis is an exposition of the history and devel- opment of indecomposable continua theory from its origins in 1910 until the present. It traces the rise of indecom- posable continua from the status of pathological examples to that of a general body of knowledge playing a fairly important role in point-set topology. The theory of ordinary indecomposable continua is explored in great detail. In addition, most of the results arising from the study of various special cases of indecom- posability are surveyed. However, no results concerning generalized indecomposable continua are included. Chapter 2 gives some background material from general topology. The specialized definitions are introduced as they are needed. Chapter 3 presents some early examples of indecompos- able continua in essentially the same terminology as the inventors used. Most of the results of Chapter 4 are structure theorems dating from the 1920's; many are still important today. In Chapter 5, several relationships Francis Leon Jones between indecomposability and irreducibility are explored. Chapter 6 presents some more examples of indecomposable con- tinua and outlines Knaster's construction of a hereditarily indecomposable continuum. Chapter 7 deals with some existence questions. In particular, the theorem that every metric space of dimen- sion greater than one contains an indecomposable continuum is proved. A proof is outlined showing that most plane con- tinua are hereditarily indecomposable. Bellamy' non- metrizable indecomposable continuum is also included. Chapter 8 presents Kuratowski's common boundary theo- rem for E2 and several of Knaster's examples. Chapter 9 relates accessibility to indecomposability. Chapter 10 treats topological groups and inverse limits. Wallace's work on clans constitutes the first part of the chapter, while inverse limits and solenoids make up the last. Chapter 11 examines the results of subjecting indecomposable continua to several usual topological operations. Chapter 12 surveys the work from Moise's thesis in 1948 to the present. The pseudo—arc and pseudo-circle are discussed, along with theorems for ordinary hereditarily indecomposable continua. A HISTORY AND DEVELOPMENT OF INDECOMPOSABLE CONTINUA THEORY By Francis Leon Jones A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1971 ci 7/ 782 Q/Copyright by Francis Leon Jones 71 For Pat and Doug, Dad and Mom, Dad and Mom Brumfiel ii ACKNOWLEDGEMENTS I wish to thank Professor J. G. Hocking for suggesting this topic and for his help and enthusiasm throught the course of this investigation. Special thanks are also due to Professors H. S. Davis and G. W. McCollum for the many interesting and helpful conversations we have had about this topic. I also wish to thank Professors P. H. Doyle, C. Martin, C. Seebeck, and R. Spira for serving on my guidance commit- tee. This research was supported in part by National Science Foundation Grant GU 2648. iii TABLE OF CONTENTS Chapter Page 1. Introduction . . . . . . . . . . . . . . . . . . l 2. Background Definitions and Notations . . . . . . 4 3. Early Examples . . . . . . . . . . . . . . . . . 15 4. Basic Structure Theorems . . . . . . . . . . . . 31 5. Indecomposable Subcontinua of Irreducible Continua . . . . . . . . . . . . . . . . . . . . 58 6. Knaster's Thesis . . . . . . . . . . . . . . . . 78 7. Existence of Indecomposable Continua . . . . . . 87 I 8. The Common Boundary Question . . . . . . . . . . 102 i 9. Accessibility of Plane Continua . . . . . . . . 125 10. Topological Groups and Inverse Limits . . . . . 132 11. Operations on Indecomposable Continua . . . . . 158 12. Hereditarily Indecomposable Continua . . . . . . 162 189 Bibliography 0 O O O O O O O O O O O O O O O O O O 0 iv LIST OF FIGURES Figure 3.1 3.2 3.3 3.4 4.1 6.1 6.2 6.3 6.4 10.1 Brouwer's Example . . . . . . . . . . . . . Knaster's First Semi—Circle Example . . . . The Lakes of Wada (Part I) . . . . . . . . The Lakes of Wada (Part II) . . . . . . . Another Plane Indecomposable Continuum . . An Example from Knaster's Thesis (PartI) . An Example from Knaster's Thesis (Part II) Knaster's Second Semi-Circle Example . . . Knaster's Hereditarily Indecomposable continum O O O O O O O O O O O O O O O O O Knaster's Common Boundaries: B3, 03’ Coo . Vietoris' Dyadic Solenoid . . . . . . . . . Page 20 24 27 27 46 80 81 83 85 116 147 ,—_.__,*# A CHAPTER 1 INTRODUCTION This thesis is an exposition of the history and devel- opment of indecomposable continua theory from its origin in 1910 until the present. It traces the rise of indecom- posable continua from the status of pathological examples to that of a general body of knowledge playing a fairly important role in point-set topology. We shall explore the theory of ordinary indecomposable continua in some detail. We.shall also survey many re- sults arising from the study of various special cases of indecomposable continua. However, we shall not include results from the theory of generalized indecomposable con- tinua, since this vein of research has not yet been as widespread as those of the ordinary and special theories. Much of the work on generalized indecomposable continua has been done by P. M. Swingle and C. E. Burgess. In Chapter 2, we give some elementary background mate- rial from general topology. The specialized definitions we shall use later will be introduced as needed. Chapter 3 presents some early examples of indecomposable continua in essentially the same terminology as the inventors used. Most of the results of Chapter 4 are structure theorems 1 2 dating from the 1920's; many are still important today. In Chapter 5, we explore several relationships between indecom- posability and irreducibility. Chapter 6 presents some more examples of indecomposable continua and outlines Knas- ter's construction of a hereditarily indecomposable con- tinuum. Chapter 7 deals with some existence questions. In par- ticular, we show that every metric space of dimension greater than one contains an indecomposable continuum. We also outline a proof that in the space of all continua of 12, the set of hereditarily indeComposable continua is a dense G5 set. Further results of this nature are in Chap- ter 12. Bellamy's non-metrizable indecomposable continuum is also included in Chapter 7. Indecomposable continua arose from a study of common boundaries of plane domains, and Chapter 8 continues this investigation. Kuratowski's theorem for E2 and several of Knaster's examples are included. Chapter 9 relates accessi- bility to indecomposablity and gives Kuratowski's charac- terization of the latter in terms of the former. Chapter 10 treats topological groups and inverse lim_ its. Wallace's work of clans constitutes the first part of the chapter, while inverse limits and solenoids make up the last. Chapter 11 examines the results of subjecting inde- composable continua to several usual topological operations. Chapter 12 surveys the work from Moise's thesis in II 1948 to the present. The pseudo-arc and pseudo-circle are W— . “-‘Hm- ____. “a __ _ ___ _ _ _ '5 3 discussed, along with theorems for ordinary hereditarily indecomposable continua. Because this work is recent and readily available, few proofs are included. s..—v.__ CHAPTER 2 BACKGROUND DEFINITIONS AND NOTATIONS This chapter gives the fundamental definitions needed from general topology, beginning with a formal definition of a tepological space. Let X be any set. By a topology gan, we mean a col- lection T of subsets of X which satisfy the following con- ditions: (l) C and X are members of T; (2) the union of any collection of members of T is a member of T; (3) the intersection of any finite collection of members of T is a member of T. The members of T are called 2222.223E1 and X together with its topology T is a topological spagg. Where no confusion can result, X is used to denote both the underlying set of points, as well as the topological space. If xeX, a neighborhood for x is any cpen set in X containing x and will be denoted by U(x). (Some defi- nitions of neighborhood require only that it be a set con- taining an open set containing x.) A set may have many distinct topologies on it, with topologies T and T' being distinct if there is an open set -—............—. w..- . -.. 4 I, 5 in one that is not in the other. The branch of mathematics known as topologies studies the consequences of imposing a topological structure upon a set. Before giving any more definitions, we make a few remarks about the ones above. The definition of a topological space did not spring into being as the result of any one person's inspiration, but, like most abstractions, it developed as a result of many persons' work and experiments. Of course, the choice of axioms for a mathematical system is somewhat arbitrary, with the only major restrictions being consistency and com- pleteness. But to be useful, a system must neither be too general nor too restrictive; in either case, very little of value ensues. Historically, topological spaces had their origin in the process of giving analysis a rigorous foundation [84]. Several concepts from analysis were generalized and ab- stracted in this process, among them being "limit," "neigh- borhood," "continuity," and "distance". In real analysis, given the notion of distance, we may say that "x is near y" if for some real number r:rO, (x-y\¢r, and that all such x for a given positive real num- ber r constitute a neighborhood of y. If this neighborhood concept is abstracted to a more general setting, not neces- sarily involving distance, then "x is near y" can be given meaning by saying that x is in some neighborhood of y, where neighborhood has been defined, say in the manner described on page 4. ,— _~.—— - 6 Next, consider a fixed set A C E1 and some point ye E1. If for any r>O, there exists X6 A such that [x-ykr, then y is a limit point of A c El. Certainly this concept can be defined in terms of neighborhoods, with no reference to distance. On the other hand, given the concept of limit (or cluster) point, neither of the other two notions can be defined in terms of it. This "linear ordering" of the three concepts was known as early as 1914, when Hausdorff noted it in his Grundzuge d2; Mengenlehre. He used neighborhood axioms to define a topological space, but it was recognized later that the "open set" axioms (p. 4) are simpler. Once topological spaces have been defined, a precise definition of continuous functions can be given. For exam- ple, f: X-—)Y is continuous at erIX iff for each neigh- borhood V in Y of f(xo), there exists a neighborhood U in X of X0 such that f(U) C'V. f is continuous 22.; if it is 1 continuous at x, for each xeX. Alternatively, f is con- tinuous on X iff for each open V in Y, f'1(V) =-CerXI f(x)6“V} is open in X. Continuity can also be specified in terms of "closed sets", which we do, following these definitions. A C X is closed iff X-A is open in X. These sets can be described in other ways. For example, if A c X, the set of cluster points of A is A' = {xexlv U(x): U(x) n (A- x ) # 95}. ‘The closure of A is A = AIJA', and A is closed iff A = I. If A is closed and A s A', then A is perfect. 7 The open sets can be described in different ways, too. The interior of A czx is Int (A) = x-XZI, and A is open iff A = Int (A). The boundary of A CIX is Fr(A) = AITX:I. In terms of the above definitions, continuous functions can be characterized by the property that for each A CIX, f(K)‘c 'TTKT. Furthermore, f is a homeomorphism iff f is one-one, onto, and for each A CIX, f(I) = f(A); that is, iff f is one-one, onto, and both f and f"1 are continuous. We shall need a few more basic definitions. A c X is genes in x if I': x. A is nowhere ESEEE in X if A c:§:I.' A collection,B,of subsets of X is a basis for a topology T on X if each member of T is the union of members of B. If B is countable, then the space is said to be 22_countab1e. Thus, to specify a particular topology T for a set X, we need not specify all the open sets; we can describe a "smaller" collection of open sets and still have the origi- nal topology. In this thesis, we shall be primarily interested in certain special t0pologica1 spaces, such as Hausdorff or metric spaces. A metric d on a set x is a function a: Xxx—49El sat— isfying: (l) d(x,y) 7/ O, for all x,yé X; (2) d(x,y) (3) d(x,y) = d(y,x), for all x,yeX; 0 iff X = y; (4) d(X9Y) .4. d(x,z) + d(z,y), for all x,y,z EX. B(X.I‘) = {ye X Id(x,y) < is 0 t I 14 disconnected, then there exist disjoint, nonempty sets K1, K2 that are closed in I; Co, and hence are closed in Co“. Then K1 and K2 are compact, and, since Co“ is T2, there exist disjoint sets 01, 02 open in C 0!. with Kl C 01 and K2 C 02. . O1 U02 :3 n“ Go, implies that (C°.V . . and 33?: En = O. . be a sequence of O O 27 "On the first day a canal is built from the lake such that it does not meet the sea.water and such that the shortest distance from any point on the shore of the sea to that of the lake and canal does not exceed E1. The enp- point of this canal is denoted by L1. Figure 3.3 "On the second day a canal is built from the sea, never meeting the fresh water of the lake and canal constructed the day before, and the work is continued until the shortest distance from any point on the shore of the lake and canal filled with fresh water to that of the sea and canal filled with salt water does not exceed E . The endpoint of this canal is denoted by S2. [See Fig. 3.4.] Figure 3.4, "On the third day the work is begun from L1 never cutting the canals already built, and the work is continued until the shortest distance from any point on the shore of 28 the sea and canal filled with salt water to that of the lake and canal filled with fresh water does not exceed E3. The endpoint of this canal is denoted by L3. "Now it is clear that we can continue the work day by day in the above way, by adequately narrowing the breadth of the canals, since the land is always semi-continuous [i.e. a semi-continuum] at the end of the work of every day. If we proceed in this way indefinitely, we get at last an everywhere dense set of waters, fresh and salt, which never mingle together at any place. "Now denote by ML the shore of the lake and canal filled with fresh water, and by Ms that of the sea and canal filled with salt water, and by MP the set of limiting points of ML and Ms not contained in them. Then the sum of ML' MS, MP forms a continuous set [continuum], and any three points, each taken from the above different sets form a system of three points, every two of which form a pair of principal points of the set [i.e. the continuum is irreduc- ible between any two of those three points]. ". . . If we suppose that there are many such lakes in the land, we may obtain by the similar method a continuous set having the property [131, pp. 60-62]." The construction mentioned in the last paragraph is carried out in Hocking and Young's TOpology [44, pp. 143- 144] for two lakes. Yoneyama supplies no further proof that his set has the desired properties, which is fairly typical of the era prior to 1920. Parenthetically, it is interesting to note that other new disciplines were studying pathological examples of their own. In the same volume of the T8hoku Mathematical Journal in which Yoneyama described the Lakes of Wada, Sierpinski gave an example of a non-measurable set which is a slight generalization of today's standard example [110]. A further investigation of the Lakes of Wada was made by Paul Urysohn [116, pp. 231-233] as a tool in his monu- mental study of Cantor manifolds in a separable metric 29 space [115], [116], [117]. The goal of his work was to establish the most general possible topological definitions of "line" and "surface". Much of this work was published posthumously under the supervision of Paul Alexandroff, following Urysohn's death in 1924. His untimely death at the age of twenty-six was the result of a swimming accident [1]. Urysohn's contribution to the Lakes of Wada was an outline of a proof of the indecomposability of the con- tinuum, based on a necessary and sufficient condition for indecomposability which he had developed. (See Chapter 5.) He noted that for "a convenient distribution of canals" the continuum is indecomposable, but that he did not know if the "construction always gives an indecomposable continuum for any distribution of canals [116, p. 232]." He also indicated that the construction can be gen- eralized by: 1) allowing a countable number of lakes, provided that they "converge to a single point"; 2) allowing certain lakes or even all lakes to have no canals; 3) allowing other lakes to have several, or even a countable number of canals; 4) allowing certain canals to have only a finite length [116, p. 233]. In closing this chapter, we note one more contribution to the study of the Lakes of Wada. While trying to extend 30 Schoenflies' results on plane sets of points to higher dimensions, R. L. Wilder showed that the Wada construction does not necessarily yield an indecomposable continuum in E3 [129, pp. 275-279]. His first result was to use this construction to dig tunnels in a certain solid to get a surface which is a Peano continuum (and hence decomposable), and yet is the common boundary of three (or countably many) domains in E3. More will be said about this in Chapter 8. CHAPTER 4 BASIC STRUCTURE THEOREMS Prior to 1920, there were only two papers on indecom- posable continua which could be considered theoretical. The first was by Arnaud Denjoy in 1910 [26], and the second was Yoneyama's in 1917 [131]. However, neither work seems to have been very influential in the study of indecompos- able continua. In his paper, Denjoy announced that he believed "one could construct three domains and even a countable number of domains which all have the same boundary [26, p.28]." In such a case, "the points of such a frontier F situated on an arbitrary straight line must form a perfect every- where non-dense [nowhere dense] set e if the line contains no continuous [connected] portion of F [26, p. 138]." Since Denjoy's results were stated without proof, and later papers make scant reference to them, we say no more about them. Yoneyama's paper had only slightly more impact on later theoretical investigations of indecomposable continua. One of his theorems was used by Kuratowski to help establish a theorem on indecomposable continua [69, p. 208]. One reason why Yoneyama's work does not seem to have 31 F“ 32 been very influential is that his terminology was highly non-standard with respect to the European school of mathe- matics. For example, he used the word "component" to mean subset, and "continuous set", which corresponds to our word "continuum", to mean a connected perfect set. Furthermore, in place of "irreducible continuum", Yoneyama's concept was stated in the following terms. Let S be a "continuous set" and let a, b be points of S. a and b are principal points of S if no proper "continuous component" of 8 contains a and b [131, p. 47]. It is interesting to compare this terminology to the European, so we state a theorem of Yoneyama both ways. The original version reads as follows: "When a continuous set has two pairs of principal points, it has always two pairs of them having one point in common [131, p. 48]." On the other hand, Kuratowski, in using the above result, stated it this way: "If e is irreducible between a and b and bet— ween c and d then e is irreducible between a and c or a and d [69, p. 208]." One apparent difference in the above viewpoints seems to be that Yoneyama classifies "continuous sets" according to the number and type of principal points which they pos- sessed. 0n the other hand, Kuratowski and other Europeans studied the entire continuum, rather than just certain' points of it. Their technique seems a little more natural in the sense that irreducibility between two points of a set results from the structure of the set, rather than from 33 any property inherent in those points alone. As we proceed into more specialized concepts, we find the differences in terminology and technique growing. Yoneyama‘s definition of what we would call an indecompos- able continuum is given as: "a continuous set having a system of three points, every two of which form a pair of principal points of the set, is called a singular set of points [131, p. 62]." He proved several theorems concerning some properties of singular sets, but he gave no necessary and sufficient conditions for a continuous set to be singular. Because he was interested primarily in the principal points of a set, rather than in properties of the entire set, it is doubtful that he knew or was interested in the fact that singular sets are indecomposable. For example, his only use of the Lakes of Wada was to show that singular sets exist. Thus, a second reason why his work does not seem to have had a significant influence is that his point of view and direc- tion diverged from those of his western contemporaries. Beginning in the early 1920's, indecomposable continua were studied more as entities in themselves, rather than just as pathological examples. The first European paper devoted exclusively to studying properties of indecompos- able continua -sans examples —-was published by the Polish mathematician Stefan Mazurkiewicz in 1920. In it he an- swered affirmatively the following question posed by Janis- zewski, Knaster and Kuratowski [86, p. 35]. Given an 34 indecomposable continuum C, can one determine two points in C such that C is irreducible between them? In fact, he proved a stronger result. Using Baire category theory, he was able to show that an indecomposable continuum in En has three points such that the continuum is irreducible between any two of them. Moreover, it appears that he was the first to use the word "indecomposable" to name these sets, and R. L. Moore credits him with being the originator of the term [100, p. 363]. Instead of giving Mazurkiewicz' results here in more detail, we include them in the next section of the chapter where they can be more naturally presented. It should be noted here that during this time, the word "continuum" meant a closed connected set rather than a compact connected set. However, Mazurkiewicz restricted his work to bounded, closed, connected sets in En. So, thanks to the Heine—Borel theorem, his concept of continuum coincides with ours in En. The fact that he worked in En was not a restriction as far as his contemporaries were concerned, since they too were working in Euclidean space. Often, papers of this era made no explicit mention of what their underlying space was, perhaps because the geometric nature of the results and examples seemed to be self-evident. Perhaps also, interest in more general spaces had not yet become widespread. By far the most significant paper published on indecom- posable continua theory during the early 1920's was "Sur les Continus Indecomposables", by Janiszewski and Kuratowski. 35 It appeared in the first volume of the Fundamenta Mathe- maticae (1920), the same one which contained Mazurkiewicz' above mentioned paper. The importance of the Janiszewski and Kuratowski paper lies in the fact that it gave several necessary and sufficient conditions for a continuum to be indecomposable. The authors also defined the fundamental concept of a "composant" and established some properties of such sets. The significance of this paper is best proved by the many later references to its results. From the fact that several proofs make explicit use of the metric properties of Euclidean space, it seems likely that Janiszewski and Kuratowski considered an indecomposable continuum as a subset of some En. However, their defi- nitions and results can be placed in a more general setting very easily, and we shall do just this. Before discussing any of the results, we present some definitions. A set A is called a boundary set in X if A C X-A. A subcontinuum K of a continuum C is called a continuum of condensation if K C C-K. Hence, if K is a continuum of condensation of C, it is nowhere dense in C, since it is closed. Equivalently, K is a continuum of con— densation of 0 iff 531? = c. For if K is a continuum of condensation of c, then c = KU (0-K) -.-.- KU (‘07:?) = 621?; the converse is trivial. These definitions are still used today, with the only change being that the word "continuum" generally means a compact connected set rather than a closed connected set. 36 The first three results were not part of the original paper, but we include them here because they simplify some of Janiszewski and Kuratowski's proofs. Lemma 4.1: A subset Y of a space 0 is connected iff there do not exist two nonempty subsets A, B of Y such that Y = ALJB and such that (KllB) u (ArfB) = ¢ [44, p. 15]. 3393:: If the sets do exist, then C-I’is an open set containing B and C4B is an open set containing A. Therefore Y = [(c-I)I1Y] u [(c4§)rwy], [(o-I)r1Y] n [(cié)riy] = [(CJI)rI(c£§)] n Y e = [c-(KUEH n Y (6. and Thus Y is disconnected. If Y is disconnected, then Y = (OtlY) U (VilY), where O, V are open in C, and Ole, Vle are nonempty and disjoint. Set A = Ole, B = Vle. Lemma 4.2: Let X be a connected subset of a connected set C. If C-X is disconnected, say C-X = MlJN, then XlJM and XtJN are connected. Moreover, if X is closed, then XlJM and XlJN are closed [62, pp. 210-211]. 3339;: By Lemma 4.1, c-x = MlJN, where M i ¢ s N, and (MITF) U (fiWlN) = ¢. Suppose XtJM = AlJB, where A i ¢ and B £ ¢. and (KIIB) u (AfIB) = ¢. Since x is connected, we may assume XIlA = ¢, whence A C‘M. We now disconnect C. C==XUMUN=AJMBUNM A # ¢ £ (BlJN); (AnB)U(Anfi)c(AnB)U(Mnfi)=¢; (EnB)u(XnN)c(1nB)u(finN) ll An(BUN) ll ‘8 Kn(BUN) 37 This contradiction establishes the first part of the Lemma. If x is closed, then m = XuM = xuM = (XUM) n (XUMUN) = XUM, since MnN = 95. Therefore, XUM is closed. Likewise XLJN is closed. Lemma 4.3: Let C be an indecomposable continuum and let K be any proper subcontinuum. Then C-K is connected. 2323;: If C-K is disconnected, then C-K = MlJN, where M, N are nonempty sets such that (MTIN) U (Mle) = C, by Lemma 4.1. By Lemma 4.2, KlJM, KlJN are continua, and their union is C. Since each is a proper subcontinuum, we have a contradiction. The next theorem is of major importance. It was included in the Janiszewski and Kuratowski paper, and is due entirely to Janiszewski [50, p. 210]. Theorem 4.4: In order that a T2 continuum 0 should be indecomposable, it is necessary and sufficient that each proper subcontinuum of C should be a continuum of conden- sation [50, p. 212]. Proof: If C is decomposable, then C = CllJCZ, where 01 and 02 are proper subcontinua of C. Thus, C—Cl C 02, from which U-Cl c:02 # C. Therefore, 01 is a prOper subcon- tinuum of C that is not a continuum of condensation of C. (This part of the proof is essentially as Janiszewski gave it.) Conversely, let C be indecomposable. Suppose that the condition does not hold, so that there exists a proper sub- 38 continuum K of c such that 611! ,é o. c = KU (C-K) = KU (62K). By Lemma 4.3, C-K is connected, and hence so is C3K. Thus, the latter set is a proper subcontinuum of C, and we have contradicted the indecomposability of C. Corollary 4.5: Let C be a non-degenerate T2 indecomposable continuum. Then C is not locally connected at any point. am: On the contrary, suppose there is a point ac—C such that C is locally connected at a. There exists bEEC, distinct from a, and there exist disjoint open sets U, V containing a, b respectively. By local connectiviety, there exists a connected open set K containing a and contained in U. K is a subcontinuum of C, and Kflv z ¢ implies K £ C. Moreover, Ktl(C-K) = 0 implies Kflajf = ¢. Thus, K is a proper subcontinuum of C which is not a continuum of conden- sation, contradicting Theorem 4.4. The converse to the Corollary is false. (Neither this statement nor the Corollary were part of the Janiszewski- Kuratowski paper.) To see that it is false, consider the continuum in E3 constructed as follows: Construct the Cantor set on the x-axis between (1,0,0) and (—l,0,0) and on the line segment joining (0,0,1) and (0,1,0). Next, construct all the line segments determined by the points of the two Cantor sets. The set so formed is a decomposable continuum that is locally connected at no point. By definition, an indecomposable continuum is not the - union of any two proper subcontinua. Surprisingly, "two" can be replaced by a "countable number", provided 0 is T2. 39 We prove this by using Theorem 4.4 and the following lemmas. Lemma 4.6: If a topological space is compact and Hausdorff, then it is regular. 3355);: See Dugundji, [28, p.223]. Lemma 4.7: Let X be regular, xex, and let U be any neigh- borhood of x in X. Then there is a neighborhood 0 of-x such that er C: U C: U. 13322;: See [28, p. 141]. Lemma 4.8: Let C be a T2 continuum. Then 0 is not the union of a countable number of closed nowhere dense subsets. Proof: Let {All}?ID be a collection of closed nowhere (:0 co dense subsets of C, and suppose that C = U A1. Then L =( 00 0 (C-Ai) z ¢, which we shall show is false. t=' Since each Ai is closed and nowhere dense in C, C-Ai is open and dense in C, for each natural number i. We shall show that n (C-Ai) is dense in 0. Suppose U is any nonempty open set in C. Then, for each i, Ur1(C-Ai) i ¢. Hence, Uf1(C-Al) is nonempty and open in C. If x is any element of this set, then by Lemmas 4.6 and 4.7, there exists an open subset Bl such that x e 131 c '31 c U n (C-Al). Likewise, there exists an open set B2 in C such that ¢ £ 82 C 32 C Inductively, we obtain a sequence {Bnk'of nonempty cpen sets such that?n C Bn_lfl(C-An), for each n. Since h on Since 0 Bi = Bk # C and C is compact, then n Eh # ¢. A.“ 0:! 4o 31 c: U n (C-Al) and En c: (C-An) n 13m1 imply that ¢£3oBczun°° °° n=‘ n £L.(C'An)’ Therefore, 2..(C'An) is dense 09 and certainly not empty. Therefore, U An fi c, 03' Theorem 4.9: A Hausdorff indecomposable continuum is not the union of any countable collection of proper subcontinua. Proof: If (K1}f' is any family of proper subcontinua t:| of C, then by Theorem 4.4, each K1 is nowhere dense in C. Each Ki is closed, so by Lemma 4.8, C £ g: K1. This result was used by Urysohn in a paper [116, p. 243] that we shall consider in the next chapter. To help establish the rest of their results, Janis- zewski and Kuratowski made the following important defi- nition. The notation is theirs. Let C be a continuum, and let aéc. P(a,C) = {c €Cla,c can be joined by a proper sub- continuum of C} . Hence, P(a,C) = £69K“ , where a 6K“ and K0, is a proper subcontinuum of C. Clearly, P(a,C) is a semi-continuum. P(a,C) is called the comppsant of a in C. In their paper, Janiszewski and Kuratowski only used the word "composant" when the above sets had the property that for all a, b in C, P(a,C) = P(b,C), or else P(a,C)r1P(b,C) = ¢. Current usage is largely as we have given it, although in some cases "subcontinuum" is replaced by "closed connected set"o The next theorem was actually proved nearer to the end of the Janiszewski- Kuratowski paper, but we shall make use of it earlier. Theorem 4.10: If a and b are any two points of an indecom- 41 posable continuum C, then the composants are either disjoint or coincident [50, pp. 217-218]. 2223;: Suppose instead that P(a,C) £ P(b,C) and that P(a,C) nP(b,C) ,é 95. Pick c eP(a,C)-P(b,C), and choose d€P(a,C) nP(b,C). By definition of composant, there exists a proper subcontinuum C1 of C such that a,c 6 01' Likewise, there exist proper subcontinua 02 and C3 of C containing a,d and b,d respectively. 01' 02' C3 being compact imply that Cl U 02 U C3 is compact. d E 02 n C3 and a e 01 n 02 imply that CltJCZUC3 is connected. Therefore, this union is a continuum, and b,c €01 UC‘2 UC3 C 0. Since c$P(b,C), C is irreducible between b and c. Thus, ClUCZUC3 = C. Finally, C = CllJCZ, or else 0 = (CIUCZ)UC3 show that C is decomposable. Using the definition of "composant", Janiszewski and Kuratowski restated Mazurkiewicz' theorems as follows: Theorem 4.11: (a) If a is any point of a metric indecomposable con- tinuum C, then the set P(a,C) is of first category. (b) For any point a in a metric indecomposable con- tinuum C, the set P(a,C) is a boundary set in C. (c) If C is a metric indecomposable continuum, then there exist three points such that C is irreducible between any two of them [50, p. 215]. M: (a): (adapted from [44, p. 140]) Let aeC be arbitrary, and let P(a,C) be as above. C being metric 42 implies that C- {a} is open and has a countable basis {on} [28, p. 233]. Let Kn(a) be the component of 0-?5'n containing a. ThenK (a) is connected, and, since C is compact, K (a) is compact. Moreover, this subcontinuum is proper, since Kn(a) C C-Un implies that Knla) C C-Un C C-Un :4 0. Therefore, K337 C P(a,C), for each n, so that we have if“ W e mac). 0n the other hand, if x6 P(a,C), then there exists a proper subcontinuum C. of C such that 0' contains a and x. Let pé C-C'; 0-0. is open in C- a , and p ,4 a shows that p6 C-{a}. Therefore, p eon, for some n such that On C C-C'. Since an C C-C', 0-011 2 0'. Then by definition of 18(9), we have C' C %(a) C C-Un. Therefore, xeP(a,C) implies m that xegl(a). Hence, P(a,C) C U“ KnIa). By Theorem 4.4. each X (a) is nowhere dense in C, and thus P(a,C) is a first category set. w (b): By (a), P(a,C) = U“ Knta), so we have C-P(a,C) = a: - - d 1 cc a is 2:‘(C K 4a)). Each C Knla) is open, an , s n Kn( ) nowhere dense, each is dense. By the proof of Lemma 4.8, m n (C-K (a!) is dense. But then C-P(a,C) being dense shows “3' that P(a,C) C C = C-Pla,C). (c): By (a), each P(a,C) is the union of a countable number of closed nowhere dense sets. If there were only a 43 countable number of composants in C, then C would be the union of a countable number of closed nowhere dense sub- sets, violating Lemma 4.8. Therefore, C has uncountably many composants. By Theorem 4.10, the composants are dis- joint, so choose exactly one point from each one. By the definition of "composant", C is irreducible between any two of these points. The above proof shows that a metric indecomposable continuum has uncountably many composants, and that it is irreducible between each two points of a certain uncount- able set. Mazurkiewicz used the same technique in Euclidean space, although he seemed to be satisfied with talking about three points instead of uncountably many. This may have been motivated by the fact that the converse needs only three points. He may have been aware of this, since Janis- zewski and Kuratowski established the converse as well as suggesting the original problem to him. Mazurkiewicz later showed [91] that a metric indecomposable continuum has as many composants as there are real numbers. Using Theorem 4.11, Janiszewski and Kuratowski were able to establish the following necessary and sufficient conditions for a continuum to be indecomposable. Theorem 4.12: The following are equivalent: (a) A metric continuum is indecomposable. (b) For each a6 0, there is a point x6 C such that C is irreducible between a and x. (c) There exists a 6C such that P(a,C) is a boundary 44 set in c; that is, P(a,C) c UIPTEIUT. (d) There exist three points of C such that C is irreducible between any two of them [50, p. 215]. 2222f: That (a) implies (b), (c),(d) follows from Theorem 4.11 (a), (b), (c) respectively. Conversely, suppose that C is decomposable; that is assume that C = CllJC2, where 01’ 02 are proper subcontinua. To establish that (b) is now false, first choose a6‘lelCZ. Then Cl C P(a,C) and 02 C P(a,C). Therefore, we have that C = ClUC2 C P(a,C), so that C is not irreducible between a and any other point of C. Statement (c) is also false now. Let as C be such that (0) holds, and without loss of generality, suppose 8.601. Since 01 c P(a,C), then C-P(a,C) c C-01 c 02. 02 being closed shows that C:P(5:CT C C2' If (c) were true, then P(a,C) C CZPTETCT would imply that C1 C P(a,C) C C:TW§:CT C C Thus, we would have Cl C 02’ which would imply that 2. C = CllJC2 = 02' This contradicts C £ C2. Therefore, P(a,C) ¢ C:PTETCT, for any a6 C. Thus, (0) is false. Finally, let a, b, c be any three points of C. With- out loss of generality, a, be 01' Therefore, C is not irreducible between a, b. This shows (d) is false. Corollary 4.13: Let C be a metric continuum. C is indecom- posable iff it is irreducible between some point p5 C and each de D, where D is a dense subset of C. 45 2522:: If C is indecomposable, then by Theorem 4.11 (b), C = C:P(§TCT. Conversely, if D is any dense subset of C, and if C is irreducible between some point p and all points of D, then C-P(p,C) D D. Thus, C:PTETCTID D = C, so that P(p,C) C C:PTETCT. By Theorem 4.12 (c), C is indecompos- able. This result was used often in the literature, but it was never explicitly stated nor proved, possibly because the proof is not difficult. As an interesting application, Theorem 4.12 (d) can be used to construct an indecomposable continuum in the manner discussed in Hocking and Young's Topology, [44, p. 142]. This example was not part of the Janiszewski-Kuratowski paper. Let pl, p2, p3 be any three distinct points of E2. Construct C1, a finite simple chain of connected open sets from pl to p3, containing p2, as shown: 46 Inside 01’ construct another finite chain of open con- nected sets, 02’ from p2 to p3 containing p1. Inside 02’ construct another such chain C3 from pl to p2 containing p3, as shown in Figure 4.1. Figure 4.1 In general, C3n+l is a chain from pl to p3 containing p2, C3n+2 is a chain from p2 to p3 containing pl, C3n+3 is a chain from pl to p2 containing p3, and for all k, we have Ck C Ck+l' 47 no no h = = - Note t at C 2:.C3n+1 gu’C3n+2 — fl. C3n+3‘ Moreover W o o o w o 2:.C3n+l is irreduchble between p1 and p3; Ll“ C3n+2 is as no C3n+3 is irreducible bet- irreducible between p2 and p3; I): ween p1 and p2. Therefore, C is indecomposable by Theorem 4.12. Although we have explicitly given only three points, p1, p2, p3, such that C is irreducible between any two of them, it follows from the proof of Theorem 4.11 (c) that C actually has uncountably many such points. (In more current terminology, the continuum C is said to be cellular, since it is the monotone intersection of a countable number of 2-cells.) Janiszewski and Kuratowski established a further char- acterization of indecomposability in terms of composants: Theorem 4.14: In order that a metric continuum C should be indecomposable, it is necessary and sufficient that it con- tains two disjoint composants [50, p. 219]. 2332:: By Theorems 4.10 and 4.11, a metric indecompos- able continuum has uncountably many disjoint composants, and hence certainly has two. Conversely, suppose P(a,C) and P(b,C) are two dis- joint composants of C. Assume C is decomposable: C = CllJCZ, where Cl and C2 are proper subcontinua of C. Either Cl C P(a,C), or else 02 C P(a,C). But, in any case, we have C1002 C P(a,C). Likewise, 01002 C P(b,C). There- fore, ¢ ¢ ClnC2 C P(a,C)rlP(b,C), contradicting the hypothesis of disjointness. Hence, C is indecomposable, and the theorem is established. In the last three pages of their monumental paper, Janiszewski and Kuratowski considered bounded closed con- nected sets, while the rest of their results held (at least in E2) regardless of boundedness. The principal theorem established in this section may be stated in our terminology as follows: "Each composant of a T2 continuum is dense." Their proof was done via metric properties, but we give a more general argument. Definition: Let X be a topological space. Define a relation "Aw" on X by Xaly iff there is no decomposition of X into two nonempty disjoint, open subsets, one of which contains x and the other of which contains y. It can easily be seen that "ha" is an equivalence relation. The equivalence classes are called the SEEEE' components of X, and we denote the quasi-component con- taining XGLX by Q(x). Moreover, Q(x) is the intersection of all closed open subsets of X containing x [76, p. 148]. Furthermore, the component of x, C(x), is contained in Q(x). For if A is any closed open set containing x, then C(x) C A, and hence C(x) C Q(x). To see that C(x) C A holds, suppose C(x) ¢ A; then C(x) = AlJ[C(x)-A], contradicting the con- nectedness of C(x). Lemma 4.15: In a compact T2 space, the components coincide with the quasi-components. Proof: By the above remarks, we have C(x) C Q(x), for 49 each x6 X. By the maximality of C(x), it suffices to show that Q(x) is connected in order to establish the opposite inclusion. Suppose Q(x) = AlJB, where A, B are nonempty, disjoint, closed subsets of Q(x). Then A, B are closed in X, since Q(x) is closed in X. Therefore, A, B are compact, since X is compact. X being T2 implies there exist disjoint open sets in X, U, V, such that A C U, B C V [28, p. 225]. Let M = X-(UUV); M is closed in X. Let {EJMQ be all the closed open sets in X containing x, so that Q(x) = SHE“. Now,2( (M0F“)=an(1ro, =Mn(AUB)=¢. By applying De Morgan's laws to the definition of compactness, it follows that there exists [me] n (:l D such that n(M11£g.) = C:: " n ¢. Therefore, 0, F“, C UlJV. L‘ L '\ Claim: (0 Fu- )llU is closed open in X. —-_' \': I L ~33 F '\ ~I a, is certainly closed open and (?fl F“, )rlU is A _ . clearly open. Moreover, (0~ F“; )[1U 15 closed, and L" Q n _ n Q F )nU= NF,“- nU), since 9:,ch cqu and Unv: t | ( q; ¢. Therefore, the claim holds. '\ There exists we Q(x) such that we QafFfit flU,s1nce A u I l ¢ £ A C (Qn Fu; ) n U. ¢ # B implies that there ex1sts a n n zeQ(x) such that z eX-f] F“; 0U. But then t]:- Fou. nU and L” its complement are nonempty, disjoint, open subsets of X, one of which contains w and the other of which contains 2, contradicting the fact that Q(x) is a quasi-component. 50 Therefore, Q(x) is connected, and hence Q(x) C C(x). We use these results to establish the following impor- tant lemma. Janiszewski and Kuratowski also established it, although they did not use the "quasi-component technique". Lemma 4.16: If K is a proper subcontinuum of a T2 continuum C, then there exists a subcontinuum L such that K'E L CFC [50, p. 220]. 2322;: From C being compact and T2, it follows that C is regular. Then x5 C-K and K closed imply there exists an open V such that x 6V and VnK = C. Therefore, K C C-V, and 6:? # C. Let L be the component of 6:? containing K. L is connected , closed in the closed set 0-7 and hence also closed in the compact set C. Thus, L is a proper subcon- tinuum of C. It remains to show that L £K. 0n the one hand, K n C-C-V = 0, because ¢ = Kfl(C-C-V)== K n C-C-V : K n c-c-V. 0n the other hand, suppose that LflC-C-V = C. Since C is compact and T2, C-V is compact and T Therefore, L is a quasi-component and hence is the 2. intersection of all closed open sets in C-V containing a given point ye K, by Lemma 4.15. Thus, L = $68G? , G9 closed open in C-V, and y<.GG . By assumption, we have (nQGp) n 0-5:? = p, whence “(.(Gfi 00-63) = 95. As before, mm m‘ ’— there exists a set [05].»; such that Q G9; nc-c-V = p. am _ ’m .. In such a case, C = n GP' U [(C-V - T Gp. ) U C-C-V], \ L lv 51 which contradicts the connectivity of 0. Therefore, we have Ln 0-53, and hence L ,é K. The above proof is patterned after several in volume two of Kuratowski's Topology [76]. Theorem 4.17: In a T2 continuum, each composant is dense [50, p. 221]. ‘Egggf: PTETCT is a continuum, since it is connected and closed in the compact set C. If PTEICT # C, then it is a proper subcontinuum of C containing a. Therefore, P a, C P(a,C), whence P(a,C) is closed and is therefore a continuum. But, since P(a,C) is then a proper subcon- tinuum of C, there exists a prOper subcontinuum K of C properly containing P(a,C), by Lemma 4.16. However, by definition of P(a,C), K C P(a,C). Therefore, P(a,C) is dense. Corollary 4.18: In a T2 continuum, a composant is not a proper subcontinuum. Proof: This was shown in the proof of Theorem 4.17. Corollary 4.19: If C is a metric indecomposable continuum, then for any aeC, P(a,C) = C; for any a e C, P(a,C) C C-P4a,0$ = C. Proof: Theorem 4.17 establishes the first statement, and the second follows from Theorem 4.11 (b). Thus, an indecomposable continuum in a metric space is "very irreducible" in the sense that given any point as C, there is a point x «:0, arbitrarily near 3, such that C is irreducible between a and x. For a given a 60, the 52 set of all such xé=C is dense in C. 0n the other hand, given any a6 C, C is not irreducible between a and all points of a dense subset. Furthermore, an indecomposable continuum is "very con- nected" in the sense that any proper subcontinuum may be removed without disconnecting it (Lemma 4.3). Knaster and Kuratowski proved [64, p. 37] a similar result which showed that any point could be removed from a (non-degenerate) indecomposable continuum without disconnecting it. R. L. Moore established an even stronger result for Hausdorff spaces along these lines. Theorem 4.20: Let C be a T2 indecomposable continuum, and let K be any proper subcontinuum. If L is any subset of K, then C-L is connected [100, p. 361]. 2322:: If C-L is not connected, then by Lemma 4.1, C-L = AlJB, where A, B are nonempty and (ArlB) U (Ale) = C. By Lemma 4.3, C-K is connected, so C-K C A and B C K. By Theorem 4.4, C = C:K C A. Therefore, K C‘A, and hence B c:K, which is a contradiction. Thus, C-L is connected. For a related result, see p. 76. Janiszewski and Kuratowski also established two other results for indecomposable continua. We shall present them, but since they play no role in our later work, we do not prove them. Theorem 4.21: Let C be a metric indecomposable continuum. Each subcontinuum situated in a composant is a boundary set with respect to that composant [50, p. 221]. 53 The following is due to Mazurkiewicz. Definition: The relative distance between x,ye S is dr (x,y) = inf8(E), where 6(E) is the diameter of E, and the infimum is taken over all connected sets E C S con- taining x,y. The relative diameter of A C S is 5&(A) = sup dr(x,y), for x,ye A. The oscillation of S at p68 is 10(p) = inffl5r(A), where A runs over all subsets of S such that pe Int (A) [87, p. 170]. Theorem 4.22: For any point of an indecomposable continuum C in a metric space, the oscillation of C at the point is a constant and equal to the diameter of C [50, p. 217]. As our final result of the chapter, we prove Knaster's first semi-circle example (see p. 24) really is an indecom- posable continuum (of p. 161). Let DO be the set of semi- circles with non-negative ordinates, centered at (l/2,0) and having as endpoints the points of the Cantor set, F. For n7/1, let Gn be as before, and let Dn be the set of semi-circles with non-positive ordinates, centered at (5/[2- 3n],0) and having as endpoints the points of Gn' Then the set B = U,Dn will be shown to be an indecomposable 0 continuum. Lets: _c8 Sn’ where Sn is an infinite sequence of semi- circles used in constructing B, satisfying: (a) (0,0) and (1,0) are in S1; (b) for nzrl, SnnSn+1 is the point of F common to both. Let K represent the points of the Cantor set,F§ which are ___— _.-L:. -o 54 endpoints of some Jn,k (see p. 9 for the notation). It can be shown by induction that K c:s; that is, for each "end- point" in F, there is a semi-circle Sm having it as an end- point. The proof that F is perfect shows (p. 12) that K is dense in F. Therefore, K c:s and K = F imply that 8’: B. S is clearly connected, and hence so is B. S is compact by the Heine-Borel theorem, so we have shown that B is a con— tinuum. We shall next show that F31? = F. Given xeK, and e 70, we must find ye F-K such that )x-yk g . Since x6 K, then b x = «—E, where bne:{0,2}, and there exists N such that 3 n3) for all n7; N, bn = 0, or else bn = 2. Choose Nl7/N so large that l/(3N‘ '1) 4/(3n°+2). Therefore, the entire boundary of P is disjoint from L. ,. 9' 31, Since L is a continuum, we either have L C P or LflP —c:~ II contradicting r e L-P and q e L n P. Therefore L C A C which establishes the lemma. To verify the irreducibility of B between (0,0) and any point of B-S, assume that there exists a proper subcon- tinuum L of B such that (0,0) L, and ye L, for some ye B-S. Then LrIS 4 ¢, and Lr1(B-s) # C, so that L ¢ 5. This contradicts the lemma. Therefore, we conclude that no such L can exist. Thus, B is irreducible between (0,0) and each point of B-S, which proves the indecomposability of B. The above proof is slightly modified from the one that appeared in a paper by Knaster and Kuratowski [64]. They were dealing with closed, connected, non-bounded sets in En. This bounded example and proof were included because they wanted to invert B-(1/2,l/2) with respect to a unit circle centered at (l/2,l/2) to obtain a closed, connected, non- bounded indecomposable set in E2. Thus, the proof of the indecomposability of Knaster's first semi-circle example appeared two years after the example itself. 57 We conclude this chapter with a few historical obser- vations. Zygmund Janiszewski made several great contri- butions to mathematics in general and to continua theory in particular [49]. His thesis established many results on irreducible continua that continue to be of use today. Of course, the above paper with Kuratowski developed many of the fundamental properties of indecomposable continua. Janiszewski was also very instrumental in establishing both the Polish school of mathematics and the journal, Funda- mgnta Mathematicae. Sadly. the first volume carried his obituary. He died January 3. 1920 at the age of 32, as a result of a long illness. The second remark concerns the Fundamenta Mathematicae itself. It was founded by Janiszewski, Mazurkiewicz, and Sierpinski to be a journal dealing with set-theoretic prob- lems written in French, English, German, or Italian. This restriction of topic did not put the journal out of print for lack of papers, as some mathematicians of that day had feared. It even survived Nazi occupation in World War II, although many of its contributors did not [78]. CHAPTER 5 INDECOMPOSABLE SUBCONTINUA OF IRREDUCIBLE CONTINUA In this chapter we shall consider indecomposability as a special case of irreducible continua theory. In partic- ular, we shall exhibit some conditions that are both neces- sary and sufficient for an irreducible continuum to be indecomposable. This will give a partial answer to the question:"How much stronger is the condition of indecompos- ability than that of irreducibility?" Moreover, the inter— relations between the two concepts will be more clearly exposed. Historically, the papers cited here date from 1922 to 1927. and all but one of them were written, at least in part, by Kuratowski. Some of the results obtained in those papers were valid only for non-bounded sets in Euclidean spaces. These are omitted, not only because such sets are not continua by our definition of a continuum, but also because they do not contribute to our later developments. The first result to be considered here was proved by Paul Urysohn [116, p. 226] in 1926. His work was done in a general metric space under the same definitions that we use today. His definition of "compact" is actually our term "countably compact", but there is no distinction between 58 59 these concepts in a metric space [28, p. 230]. It is interesting to note that this is the first paper we have discussed in which the definitions agree with current usage. Theorem 5.1: In order that a metric continuum C, irreducible between a and b, should be indecomposable, it is necessary and sufficient that it contain a semi-continuum S such that: (a) either a or b is in S; (b) S}: C; (0) C38 = C. 2322:: If C is indecomposable, then by Theorem 4.12, C is irreducible between a and some erC. Set S = P(a,C). By Theorem 4.11, 5:8 = C, and by Theorem 4.17, S = C. On the other hand, suppose the conditions of the theo- rem are satisfied, and without loss of generality, assume aé S. We claim that P(b,C) GS = C. If not, choose x in the intersection. Since xe S, there exists a continuum K C S, with a, x6 K. x€P(b,C) implies that there is a continuum K. C P(b,C), with x, bEFK'. 0 being irreducible between a and b implies c = K UK'. Therefore, c-s c C-K c K', from which it follows that c = as c: K' c: P(b,C). But then a €P(b,C), which contradicts the irreducibility of C between a and b. Thus, the claim is established. Since P(b,C)flS = C, S C C-P(b,C). Therefore, we have m a s = c. By Theorem 4.12, c is indecomposable. (This proof is essentially as Urysohn gave it [116, pp. 226- 227].) Urysohn notes that as a necessary condition, the theo- rem is not very interesting. However, it does provide pre- cisely sufficient conditions, which he shows by removing each condition one at a time and constructing counter— examples. As mentioned earlier (p. 29), he used this theorem to outline a proof of the indecomposability of the Lakes of Wada. Essentially, he lets MS play the role of S, and he states that the irreducibility of C = MSUML U MP follows from a convenient distribution of the canals [116, p. 232]. We next consider several results which Kuratowski established in [69]. This work was the major portion of his thesis, written under the direction of Mazurkiewicz and Sierpinski in 1920 [69, p. 201]. It also contained the previously discussed Knaster's "semi-circle example" (pp. 24, 53), and is seemingly the only paper to use results of Yoneyama. Using Kuratowski's notation, let 0 be a T2 continuum irreducible between a and b, and define R(a,C) to be the empty set together with the set of all subcontinua L of C containing a such that L = 6:65L. The equation simply requires that L = Tit—TL), a condition sometimes referred to by saying that L is a regular set. This is not to be confused with the separation axiom of the same name. Before we can prove any of the major theorems, we must establish some background results. Lemma 5.2: Let C be irreducible between a and b, and let K be a closed connected subset. Then C-K is either connected 61 or else it is the union of two connected sets, one of which contains a and the other of which contains b. If aEK, then C-K is connected [69, pp. 202-203]. ‘22222: Suppose C-K is not connected. Then C-K = PlJQ, where P, Q are nonempty, disjoint, open subsets of the open set C-K. By Lemma 4.2, KlJP, KlJQ are closed connected subsets of C, and hence are subcontinua. But then we have C = (KUP) U (KUQ). If aeK, then either KUP or KUQ is a proper subcontinuum of C containing a, b. This violates the irreducibility of C. Therefore, if aeK, then C-K is connected. Since a, b are not both in either KlJP or KlJQ, we may assume aeP, and b 6Q. Hence, C-(KUP) = Q is connected by the first part of this lemma, and likewise P is connected. Lemma 5.3: If A, B are two closed connected subsets of C, with C irreducible between a, b, where a6 A, b EB, then C-(AlJB) is connected [76, p. 193]. 2322:: We may assume Ale = C, for if not, then AlJB is a subcontinuum of C containing a, b. Thus, C = AlJB by the irreducibility of C, and consequently C-(AlJB) = C. C-A is connected by Lemma 5.2. Suppose that the set (C-A)-B = C-(AIJB) is disconnected. Then it is the union of two nonempty sets U, V such that (UIIV) U (UTIV) = C. By Lemma 4.2, BlJU and BlJV are connected. Hence, their closures, BlJU, BlJV are connected. Since C = (AlJB) U TC:KT:B, and AIIB = C, then AllTC:K7:B s C. For if not, then A and B11T62K72B show 0 62 is disconnected. Moreover, C C AIITC:K7:B = AIIUTTV implies that AIIU C C, or Ath C C. Without loss of generality, suppose AIIU C C. Therefore, AlJUlJB is connected and con— tains a, b. By the irreducibility of C, C = AtJUlJB. Therefore, V C (C-A)-B CIU; however, VI]U = C, so we must conclude that V = C. This contradicts the fact that V C C. Hence, C-(AlJB) is connected. Lemma 5.4: Let C be a continuum irreducible between a,b, and let K be a subcontinuum. Then Int (K) is connected [76. p. 194]. 2522;: If K = C, then the result is clear. So suppose K C C. By the irreducibility of C, not both a. b are in K; assume a 5 0-K. If C-K is connected, then so is C-K. In this case, Lemma 4.2 shows that C-C-K = Int (K) is connected. If C-K is not connected, then by Lemma 5.2, it is the union of two connected sets, P, Q with a6 P, and be Q. Let A = F and B = 6 in Lemma 5.3, and it follows that C4EIK = C-PlJQ = C-(FIJU) is connected. We need two more sequences of lemmas to enable us to establish the major results, Theorems 5.14, 5.16, 5.17. Lemma 5.5: Let S be a topological space with subsets A, B. Then I-B‘c:I:B [68. p. 183]. 23932: Let xeK-B, and let 0 be any neighborhood of x. We must show that there is a ye 0 such that ye A and y ¢ B. But, there is an open set U, x eU, such that UrlB = C. Since xe 0 CU, this intersection is nonempty and open. So ;______ .L 63 choose any yé’OflU. It is easily seen that the following formulas hold. Lemma 5.6: For any sets A, B, C: (a) (A-C)-(B-C) = (A-B)-C; (b) (A-C) U (B-C) (AlJB)-C; (c) (A-C) n (B-C) (AriB)-C; (d) (A-B)-(A-C) = A n (C-B); (e) (A-B) u (A-C) A-(Bnc); (f) (A-B) n (A-C) = A-(BLJC). Finally, we prove a sequence of lemmas which will show Int KIT? = Int K'u Int Y. Lemma 5.7: Let S be a tepological space. If X, Y are nowhere dense in S, then so is XlJY. 'Prggf: Since S:X = S = S:F, then 8-? = S:FAY C (S:F7:F, by Lemma 5.5. Thus, 3 = 5:? c:§:7§55§5'czs, from which the Lemma follows at once. Lemma 5.8: If X is nowhere dense in a topological space S, and if 0 is open in S, then Xf10 is nowhere dense in S. Proof: XIIO crx implies s-KWTC:= s-K': X 2 Xr10. Lemma 5.9: X is nowhere dense in a topological space S iff Int K = p; X is a boundary set in 3 iff Int X = C. Proof: The result is obvious from the definitions. Definition: X is locally nowhere dense (respectively a boundary set) at pars if there is a neighborhood 0 of p such that Of1X is nowhere dense (respectively, a boundary set). Lemma 5.10: X is nowhere dense at p iff X is a boundary set 64 at p. 'Prggf: If X is not nowhere dense at p, and if G is any open set containing p, then Gle is not nowhere dense. By Lemma 5.9, there is an open set H such that C C H C G7TX. Therefore, H = HFYCTTX<:'HTTG7TX, since H is open. Hence, HnG C C. Then C C HnGcGnmcGnK‘ and Lemma 5.9 show Gle is not a boundary set. Hence, X is not a boundary set at p. Conversely, if X is not a boundary set at p, and if G is any neighborhood of p, then there exists an open set H such that C C H C Gle. Since G is open, C C H C Gle C C7TX. Therefore, X is not nowhere dense at p. Lemma 5.11: Int X is the set of points of S where X is not locally a boundary set. m: p em and G any neighborhood of p imply that GnInt X C C. Then C C GnInt X c an shows that X is not a boundary set at p. On the other hand, p6 s-m implies (S-Int X) n x is a boundary set, since Int [(S-m) n X]: Int (S-Int X) n IntX (s-m) n Int x = C. Lemma 5.12: Int X’is the set of points of S where X is not locally nowhere dense. Proof: The result is immediate from Lemmas 5.10, 5.11. Lemma 5.15: Int *XUY = Int 'K' u Int Y. Proof: One inclusion is easy and requires none of the above machinery. s-EZ-K u s-fi = (8-33?) u (s-s-Y) = s-(ETK— n 33') c: 65 ____ s-[(s-K) n (s-Y)] .-. S-S-(XUX). Conversely, if p¢Int X, then by Lemma 5.12, X is locally nowhere dense at p. Therefore, there is a neigh- borhood O of p such that OIIX is nowhere dense in S. Like- wise, p¢ Int X implies there is a neighborhood U of p such that Ule is nowhere dense. Then Xf10flU and Y!]0[]U are nowhere dense in S by Lemma 5.8. By Lemma 5.7, we have (XIlOflU)lJ(Yf]OI]U) = (XlJY) n (OflU) is nowhere dense. Therefore, (XlJY) is nowhere dense at p, and hence by Lemma 5.12, pfim. The desired inclusion follows by taking complements. This last sequence of lemmas has been adapted from material in Volume I of Kuratowski's ToEology [75]. We may now return to indecomposable continua theory. Definition: Two members K1’ K2 of a family of sets 7(aform a jump [saut] if for each Kf‘Hf such that Kl C K C K2, then either K = K or else K = K2. 19 Theorem 5.14: If K is a nonempty indecomposable continuum contained in a T2 continuum C which is irreducible between a and b, then K is either a continuum of condensation or else K==C-C-K. In the latter case, there is a member R0 of R(a,C) such that R0 and ROlJK form a jump [69. pp. 210-212]. Proof: C-K C C-K implies C-C-X C'K. Then, since K is closed, C—C-K c K. By Lemma 5.4, C-C-K is connected, and hence so is C-C-K. Therefore, C-C-K is a continuum con- tained in K. If it is a proper sub-continuum of K, then, -- ="< rah '-. . . . _. .. ->-~-«.--..~-. ,4.y:4._,_ :v v” _ 66 since K is indecomposable, Theorem 4.4 requires that C-C-K be a continuum of condensation of K and hence of C. If it is not proper, then K = C-C:X. We now establish the second part of the theorem. If K = C, then R0 = C will do. For let R be any member of R(a,C), C C'R C CLJC. If R = C, there is nothing to prove. If R .4 c, then by Theorem 4.4, “diff = 0. But, ReR(a,c) implies c-‘cT-‘B = R. Therefore, R = 6273 = C. We now assume that KC C, and that a6 C-K. By Lemma 5.2, 53X = FlJQ, where Ple = C, and P, Q are open in the open set C-K. Furthermore, a (P, and either b GO or else Q = C. We shall show that in each case, R0 = F will do. We must first show that Fe R(a,C). Clearly F is a subcontinuum of C which contains a. It only remains to show F = C-C-F. As in the first part of the proof of this theo- rem, C-C-F C'F. P C'F and P open in C imply C-P = C-F D C:F. Therefore, P C C-C-F, from which the desired result follows by taking closures. The next step is to show FUK ER(a,C), and the first result needed for this is that FlJK be a continuum con- taining a. To establish this, it suffices to show that FnK C C. Since C-K = PUQ, with aEC-K and either Q = C or be Q, then C:X is a continuum containing a and b or else a and not b. In the first case, the irreducibility of C shows C = C3K. ,Therefore, F = FlJQ = C D K, and hence FflK = K C C. In the second case. F—P C C, for if not, 67 then P is an open and closed nonempty subset of C which is proper since b6 Q. This contradicts the connectivity of C. Suppose that FrlK = C. Then F C C-K. LetlceFLP; then x6 Q or xe K. But, x eK implies xfC-K, whence F C C-K. 0n the other hand, if er, then xCF, since Pr) Q = C. Therefore, BnK C C. As before, FlJK D C-C-(FLJK). For the opposite inclu- —C-F imply that tdl Q sion, K = C—C-K and FUK = (c-'C-_K) u (C-E) = c-(fi n 0-?) c C—[(C-K) n (05)] C-C-(K UF). We shall next show that F and FlJK form a jump. Let s €R(a,C) be such that '15 c s <: FUK. We must show that s = F or else S = FlJK. Since S €R(a,C), 8.68. We wish to show that S:F is connected, and this can be done by applying Lemma 5.2, provided S is irreducible between a and some other point. Thus, we first show that S is irreducible between a and all points of Fr(S). Note that Fr(S) = C iff S is closed open in C iff C is disconnected, provided C C S C C. Therefore. Fr(S) C C. To prove the irreducibility of S, let F be any subcon- tinuum of s such that a 6F, and FnFr(S) C C. We must show S C F. Ff]Fr(S) C C implies Ff](C:S n S) C C. Hence, in particular, FrlC-S C C. Moreover, by Lemma 5.2, as s 68 implies 0-8 is connected. By the irreducibility of C, bé‘C-S. Therefore, FlJC-S is a continuum containing a, b 9 and thus must be C. Consequently, C-C-S C F, from which it follows that S = C-C:S'C’F. Thus, S is irreducible between a and all points of the nonempty set Fr(S). By Lemma 5.2, S:F is connected, and hence is a subcon- tinuum of K, since S C FlJK. By Theorem 4.4, S:F is either not proper, or else it is a continuum of condensation. That is, either S-F = K, or else K-S-F = K. In the first case, FUK = Bus—:‘PcPus = s, so that s = FUK. The other case is slightly more complicated. We shall investigate it with the help of the following lemma, which will also be useful in proving the next theorem. Lemma 5.15: Let T be any topological space, let A be a sub- set such that A : T-T-K, and let B be closed in T. (a) Then A-B = A-A-K-B. (b) Moreover, if D C'A is such that D = A-A-D, then D = T-“"'T-D [68, p. 184]. Proof: (a) Since A-_A-B c A-ATB, then it follows that A-B = A-(A-A-B) D A-A-A-B. Therefore, A-B D A-A-A-B. On the other hand, A-K:F C A-(A-B) by Lemma 5.5. Consequently, A- -B CIF = B, whence A-A-A:F D A-E. (b) We need only prove that T-T-D = A-A-D. Let x be an element of T-T-D, and let 0 be any neighborhood of X in 69 T. We must show that Ofl(A-A:D) C C. That is, we must prove that there is a 260 such that zY such 87 88 that E (x.0) = f(x). and §(x,1) = g(x). for all x: x. r is nullhomotopic if it is homotopic to a constant map. Definition: Let f, g, X, Y be as above, and let A C X. Then f, g are homotopic relative pp A if there exists a con- tinuous § : XxI —>Y such that §(x,o) = f(x), E (x,l) = g(x), and §E(a,t) = f(a) = g(a), for all xe X, a eA, t éI. Definition: Let f: R——>Wn be a continuous surjection, where Mn denotes a homeomorph of In. If every continuous mapping 5: R-—->Wn which is homotopic to f relative to f-1[Fr(wn)] satisfies g(R) = Wn, then f is called essential. If f is lot essential, then it is inessential. The above terminology follows Alexandroff's "Dimension- :heorie" [3] and Nagata's Modern Dimension Theory [103]. lemma 7.1: A continuous surjection f: R-9Bn, where Bn = DCéEnllxlél} is essential iff every continuous mapping : R‘~->Bn which coincides with f on f-1(Sn_l) satisfies :(R) = Bn. 2392;: If f is essential, the conclusion follows at nce. Conversely, if the condition holds, we only need to rove that f is homotopic to g relative to f-l(Sn-l). The omotopy given by §(x,t) = t . g(x) + (1—t) ' f(x) estab- ishes this. emma 7.2: Let X be normal and A C X closed, with f:.A---9’Sn ontinuous. Then there exists a neighborhood U D A over hich f can be extended relative to Sn. Proof: See [28, p. 151] for a PTOOf Of this corollary 3 Tietze's extension theorem. 89 .emma 7.3: (Borsuk) Let X be a compact metric space, and let >C X be closed. Let f, g: D'v9Sn be homotopic. If f has n extension F: X-9Sn, then so does g, G: X—1>Sn, and C an be chosen so that F and G are homotopic. Ppppf: Let 7’be the homotopy of f and g. Define the app; : Kx{0] u DxI ——>sn by §(x,0) = F(x), §(d,t) = 1’(d,t) a extend SE to all of XxI. By Lemma 7.2, _§:has an exten- Lon a% over some neighborhood U D (Xx{Q} U Dxl). Since I 3 compact, there exists a neighborhood V D B such that :I C U [28, p. 228]. Since B and X-V are disjoint closed ets, there exists a continuous function f flC-9I, say (x) = am, if?) f‘ggx, B , such that 10(3) = 1 and f(X_V) 0. Then 32(xgt) = E§(x,t -‘F(x)) is the required homotopy. tting C(x) = 3E(x,1) completes the proof. There are also less stringent conditions on X under ich the lemma holds; for our purposes, we only need it as ated. We now present the first of Mazurkiewicz' lemmas. Lma 7.4: If f is an essential transformation from a com- :t metric space A onto B2, then f: A1 = f—l(Sl)--—,>Sl is t nullhomotopic [95, P. 327]- Proof: Suppose fIA :Al-—~9Sl is nullhomotopic. Then by I . l lma 7.3, fM can be extended to a continuous F: A.~)S ' 1 2 lich is also nullhomotopic). Then F(A) c: s C B , so by 1ma 7.1, f is inessential. l lma 7.5: Let X be any space, and let f: X-9S be con- .uous. If f(X) C 81, then f is nullhomotopic. 90 , 1 . 1 Proof. Choose s05 S -f(X), and define g: X-—>S by g(x) = —S o. to 1-t - f x . x, t = . o§( ) TM gx _ , x l is the required homotOPY. Corollary 7.6: If f is an essential transformation from a compact metric space A onto B2, then A1 = f-l(Sl) contains a component K such thatzfiSEinullhomotopic transformation of K into 81, and consequently, f(K) = 81. 2322;: Suppose that for every component K C Al, fix is nullhomotopic. (We now follow a proof of Eilenberg [29, p. 1 such that 164].) There exists a continuousf : KxI—9S §(k.0)= f'K (k) and r (k, 1) = S0 581. Let B: (A lx{0}) u (KxI) u (A lx{1)) c Al xI Set §(X,O) = f(x), §(x,l) = 80’ for all xeAl, and £(x,t) = §(X,t), for all x 6K, and t CI. There exists an open set U D B such that SF can be extended to U. As before, there is a neighborhood V D K, such that VxI C U. Since K is a Component of a compact metric space, it is a quasi-component (see P. 48). Consequently, there is a closed open set Vi I in A1 such that K c vi c V, from which leI c U. Let ‘ A §|= Vixlfisl be the extension of § on U restricted to I —- L ' l leI. §\ establishes a nullhomotopy of flK, : Vl——>S . Carry out this process for all components K to get an 0P8n covering {K;} of the compact space Al. Then there is an open subcover {K;L£:I of Al. Moreover, we may choose these sets to be pairwise disjoint, since each of the 91 finitely many sets is both open and closed. Since fl ,: v “i. ' l . . ""‘ tifis is nullhomotopic by §u£ , we define f :AlxI -) S1 *N by I (x,t) = E“. (x,T), for the unique N; such that x 6 t V4, . I is clearly continuous, f(x,0) = f(x), for all xeAl, and f (x,l)e (s “'43.. E (x,l) is not a surjection, so by Lemma 7.5, it is nullhomot0pic, say by A . Therefore, flA :Al--—)S1 is nullhomotopic by a Q(x,2t) oat 9.1/2 Y (x,t) = A(x, 2t-l) 1/2 st .91. But, f". :Al--> Sl being nullhomotopic implies, by Lemma 7.4, that f: A—>B2 is inessential. Thus, the result holds. Finally, fIA being not nullhomotopic implies f(K) = S1, by I Lemma 7.5. Emma 7.7: Let f be an essential transformation from A onto 32. Let J CIB2 be a simple closed curve, with H denoting the one of the two domains determined by J in the plane that lies in B2. Then, f is an essential transformation from f'lCi) onto B‘ = HUJ [95, p. 528]. M: H is homeomorphic to B2. For convenience, let D denote f-1(H). Suppose that flu is an inessential transfor- mation from D onto H. Then there is a homotopy i : DxI—-—>H such that § (x,0) = f(x), and § (x,l) is, say g(x), where g(x) :4 a, for all xeD and some a eH, and such that E fixes f-1(J). We extend ftp to a function F on all A by: f(x) if x 6 PD F(x) = {g(x) if x 6 D. 92 If xeDflATD, then xef_1(J): DDT-D = f-1(H)0AT-f:l—(H—) implies f(DrlKIB) = Hr]f(A-f-I(H) c Br1f7X:f:I7P5)by con- tinuity of f. The last set is contained in the set Hle§:H, which is J. Therefore, f(x) = g(x) for all x5 DrlK:D, and it follows that F is continuous. F is homotopic to f relative to f-1(Sl) by § (x,t) if xED {E (x,t) = f(x) if xé K:D. But, since g(x) C a, for all x:fI)by construction, and since flfiTB (x) C a, for all x GAID by definition of inverse image, we have that F(X) C B2. Therefore, f is inessential. Corollary 7.8: If f is an essential transformation from A 2 a simple closed curve, then A contains onto B2, with J c B a continuum K such that f(K) = J. 3322;: By Lemma 7.7, flo : D —9H is essential. Let (P: H—->B2 be a homeomorphism. We shall show that Qof‘o is an essential transformation of D onto B2. Let g be any continuous function which is homotopic, say by§ , to CFoflo , relative to (6(0le )-1(Sl) = f-1(J). It only remains to show that g(D) = B2. But, qf‘fif :DxI-49H shows that flo is homotOpic to 4?-L g relative to f-1(J). and since flu is essential, 949 g(D) = H. Therefore, s(D)== n for all nbds N(x), Nan C C, for an infinite number of Xn} . If these two sets are equal, then this set is denoted by lim Xn’ and the sequence of sets converges. we References to this concept may be found in [44, p. 100] or [75, p. 335]. It is clear that limwinf Xn C 1%m*§pp Xn' For the remainder of the discussion, 3 denotes a compact metric space. Lemma A: p>élim sup Xn iff there exists a sequence of points "““" n4»a {pnk} such that nkp P M”, n Lemma C: Every sequence of sets in S has a convergent sub- sequence of sets. Proof: See Hocking and Young [44, p. 102]. , - ' then lim X = lim X . Lemma D. If £19m.- Xn eXlStS' "H,“ nk new n 95 Proof: See Kuratowski [75, p. 339]. Theorem 7.9: Let f be an essential transformation from A onto B2, and let C C B2 be a continuum. Then there exists a continuum L C A such that f(L) = C [95, p. 328]. Proof: We first indicate a proof that there exists a sequence of simple closed curves Jn C B2 such that lim Jn = C. For each n‘wl, cover C by {B(c,l/n)}Cé C' By compact- '7», uses of C, there is a subcover {B(c(i n),l/n[}.n' . For 0 L’- each n, construct a simple closed curve Mn passing through the points C(i n)‘ Mn has a convergent subsequence-{Mnk} , by Lemma C. We claim that lim M = C. It suffices to show n-bw nk that lim inf M C C C lim sup M . From x élim inf M , n-no nk new nk new fly: it follows that x is a limit point of C, and hence is in C. If ye C, then for all nk there exist C(i,nk)6 C such that d(y,c(i,nk))4 l/nk. B(y,l/nk) nKnk C C, and it follows that every neighborhood of y meets infinitely many of the Xn's. We now prove the theorem. By Corollary 7.8, there is a continuum Kn C A such that f(Kn) = Jn’ for n = l, 2, . . . . K , so let lim K = L. [Kn] has a convergent subsequence { ni] ng, ni L C A and L is easily seen to be closed. and hence compact. L is nonempty and by a theorem in Hocking and Young [44. p. 102], L is connected. Thus, f(L) = d:29f(Kni) = lim Jni = 11111 J = C. “#00 n 96 We are now ready to establish the principal result of Mazurkiewicz' paper. Theorem 7.10: Every compact metric space of dimension greater than one contains an indecomposable continuum [95, p. 328]. 2322:: For a definition and discussion of dimension. see Hurewicz and Wallman [45] or Nagata [103]. Alexandroff established the following result [3, p. 170], the proof of which can also be found in [103, p. 59]. "A metric space A has dimension less than or equal to one iff every continuous mapping of A into B2 is inessential." Therefore, dim A 71 implies there exists a continuous function f: An—9B2 that is essential. Let CO be an indecomposable continuum contained in B2, say Knaster's first semi-circle example, shrunk sufficiently to be contained in B2. By Theorem 7.9, A contains a con- tinuum LO such that f(Lo) = C0. Knaster and Mazurkiewicz showed [65, p. 87] that L0 contains as indecomposable con- 1 such that f(Ll) = f(Lo) = 00‘ The proof is as follows. Using Zorn's lemma, it is easy to show that there tinuum L is a subcontinuum K of L0, irreducible with respect to the Property that f(K) = f(LO). If K = AlJB, where A, B are proper subcontinua, then we have L0 = f(K) = f(A)lJf(B). f(A). f(B) are subcontinua, but by the minimality of K, ' ' ' m osable. This is a neither is all of L0, whence L0 is deco p Contradiction, so the result holds, with L1 = K. This concludes Mazurkiewicz' two page paper. We shall 97 consider related questions for hereditarily indecomposable continua in Chapter 12. The existence question for non-metric indecomposable continua seems to be more difficult. This is not too sur- prising in view of the fact that the only major theorems we have presented thus far that deal with this case are 4.4 and 5.1; the rest deal with the metric case. However, in 1968, Bellamy constructed an example of a non-metric indecomposable continuum in his thesis [6]. We again need some preliminary definitions. A topological space X is completely regular if for each peX and closed set A not containing p, there exists a con- tinuous function 1’: X'—'*I such that c9(p) = l. and c[9(a) = O for all aeA [28, p. 155]. Let IX denote the set of all X continuous functions f: X-91, and let {Iflf 61 }be a X X family of unit intervals indexed by I . Let P = 7T{%f] I7élx}; its points are denoted {tf][28, p. 155]. Lemma 7.11: If X is a completely regular T2 space, then it x . __ can be embedded in PX. That is, f : x—9P given by ‘1' (x) - X {filth-1: is a homeomorphism of X and f (X) C P . Proof: See [28, p. 155]. A compactification of a space X is a pair (X,h), where X is a compact T2 space, and h is a homeomorphism of X onto a dense subset of X. The Stone-Cech compactification of X is (@(X),‘f ), where (3 (X) = W7. Lemma 7.12: For each compact space Y and each continuous ha‘ 98 f: X->Y, there is a unique continuous extension F:Q(X)—->Y (f = FP'P). Moreover, any other compactification of X having this property is homeomorphic to 3 (X). 3329:: See [28, p.243]. We now follow Bellamy, with only slight changes in notation. Let A = [1.00), and A*== @(A)-A. Actually, 4* = P (A)- f(A), but we identify A with its image in (3(A). Lemma 7.13: Let U be an open set which meets A*. Then UlJA is unbounded [6, p.30]. 2322;: Suppose UlJA is not bounded. Then UlJA C [1,x], for some xe.A. Thus, ur1(C (A)-[l,x]) is a nonempty (since C C'Un (fi (A)-A) C U[]({3(A)-[l,x]) open subset of P (A) which misses A. This is impossible, since A is dense in ((4). * I Lemma 7.14: A is a T2 continuum. Proof: For all n7’1, let An = [n, OD), and set Pn = -)(- * 00 A lJA . Then A = n | n Pn. But, Pn = A; (closure in F (A)), so that each Pn is a T2 continuum. The intersection is * . . monotone, so by Theorem 2.1, A is a T2 continuum. Note that the above two lemmas hold for any compact- ification of A, as Bellamy showed. Theorem 7.15: A* is a non-metrizable indecomposable con- ::'11uum0 P H . - * 4* = XlJY, where X, Y are proper subcontinua of A . We shall show that X is not connected. 99 Let xéX-Y, ye Y-X. Let U, v be open sets in (3 (A) such that:1) er, er, and 2) Tiny = HnY = an = C. Choose sequences {pi}:’, {qi}; , and {r1}: from A as v ' ‘ follows: Let pl 6 U nA. Choose qu pl such that ql 5 V; this is possible since by Lemma 7.13, VflA is unbounded. Next, choose r17 ql such that (ql,rl) C V; this can be done since V is open and hence qllies in some open interval in V. Suppose pk, qk, rk have been choosen for k<;n such that for each k: 1) Pk6U9 2) the interval (qk.rk) C V. 3) pk‘ qurk, and if kt n-l, then rk‘pk+l' Then, since UnA is unbounded, there exists a qn7pn such that qne V. Since V is open, rn may be chosen greater than qn such that (qn,rn) C'V. on w a The sequences {pn}‘ , {qfi} , {r5}. are all unbounded. For if not, they would have a common supremum which would have to belong to HITV, a contradiction. Define f: A-9I as follows: 0 if i is odd f(Pi) = . . . 1 if l is even 1/5 if i is odd f(qi) = 2/3 if i is even 1/3 if i is even f(ri) = 2/5 if i is odd 100 New extend f linearly to each of the intervals [pi,qi], [qi,ri], and [ri,pi+l]. Then f is a continuous function from A to I. By Lemma 7.12, f has a continuous extension F:(3 (A)—)I. F'l(0) is a closed subset of (3 (A) containing a» . . . . [p2k+l], , and hence containing all limit pOlnts of the sequence in (3 (A). Therefore, F'l(0) CA* C C. But, since the sequence lies in U, any limit point of it is an element of H, and hence does not lie in Y. Therefore, F-l(0)(]X C C, and thus 0: F(X). Likewise, 1 e F(X). But, let a.5Frl(l/3,2/3). a is a limit point of f-l(l/3,2/3) = H) (qk'rk) C'V. Therefore, a.eVfl and hence ai¢Xh Consequently, F(X)!)(l/3,2/3) = C. Then,since F is continuous and takes on the values 0 and 1, its domain can not be connected, since its range is not. The proof that A*is non-metrizable follows from a corollary [6, p. 40] which says that A* has 20 points. Thus A* can't be embedded in the Hilbert cube, as it could be if it were metrizable. The proof that A* has 20 points is rather long and will not be presented. We conclude the chapter by briefly mentioning some results dealing with a different type of existence question. Namely, how frequently do indecomposable continua occur in the space of all continua of a given space? This appears to be a rather difficult question to answer. But the sur- prising result is that "most" continua in I2 are not only indecomposable, but are also hereditarily indecomposable. In this context, "most" means that the set of hereditarily 101 indecomposable continua in 12 constitute a dense G5 set in the space of all continua of 12, when the latter set is given the Hausdorff (see [76, p. 47]) metric [94. PP. 151- 159]. In Chapter 12, we shall see that Bing established a similar result for an even more singular type of continuum (the pseudo-arc) in any Euclidean or Hilbert space. Kuratowski also notes [76, p. 202] that in any compact metric space, the set of indecomposable or hereditarily indecomposable continua are a G5 set. There does not appear to be much known about the frequency of occurrencecxf indecomposable continua in spaces other than Euclidean or Hilbert. We shall content ourselves with an outline of Mazur- kiewicz' proof that the hereditarily indecomposable con- 2 tinua in I are a dense G6 set. The nine pages of details are not difficult and the interested reader may consult the original paper for them. He defined a sequence of sets of continua in 12,{fi;]f’ as follows: [22‘ is the set of all continua C in I2 such that 0‘3 K, where K is a subcontinuum such that K = KllJKZ, and K1, K2 are continua with the preperty that gap {d(K1,p)} 7/l/n and 511:}: {d(K2,q)} é l/n. The major portion of his paper was devoted to showing that the “L's are closed nowhere dense sets. Letting n: denote the set of hered- itarily indecomposable continua in 12, F‘ the set of all 2 continua in I , and.[: the one point continua, it follows that I: = P. (I: u U ['21) Thus, the result holds. 4 CHAPTER 8 THE COMMON BOUNDARY QUESTION We have considered various examples and properties of indecomposable continua in the preceding chapters, but we have seen no application of them, except in their original role of being pathological examples. In this chapter and in Chapter 10, we shall present some other Situations in which indecomposable continua arise. The tOpic we are going to discuss in this chapter is 2 and E3 which are common bound- the structure of sets in E aries to three or more domains, which we recall is the problem that Brouwer was considering when he discovered indecomposable continua. In the plane, such common bound- aries must be indecomposable or else the union of two indecomposable continua. It seems remarkable that by shifting our setting to E3, nothing of the sort is true. In fact, there is a set in E3 which is the common boundary of three domains and is not only decomposable but is also an absolute neighborhood retract. Kuratowski and Knaster did the above mentioned work on the planar case. Most of the chapter will be devoted to their results, but we shall also mention Eilenberg's work on the common boundary question for 82, various papers on 102 103 prime end theory, and Burgess' thesis, which makes all of the above planar results into special cases of a more general theorem. In 1924, Kuratowski wrote a paper on irreducible cuts of the plane (to be defined on p. 104) in which he showed 2 into three or more domains that if a compact set cuts E and if it is the boundary of gggp of them, then the set is either an indecomposable continuum or the union of two indecomposable continua [70, p. 138]. In 1928, he was able to establish the same conclusion while only requiring the set in question to be the boundary of at least threedomains. There are no restrictions on its relationships to any other domains it may determine in E2 [72, p. 36]. In 1925, Knaster gave examples Bn’ On which cut E2 irreducibly into n domains such that each Bn is indecompos- able and each Cn is the union of two indecomposable continua. Examples of the first type were already available from the work of Brouwer and Wada. But, the existence of the second class of examples was not previously known. Thus, the "either-or" conclusion to Kuratowski's theorems can not be improved, since there are common boundaries of both types. Starting in the early 1920's, many papers dealt with the idea of cutting the plane. In particular, several papers we will present in this chapter were written using this terminology. More recently, the idea of a set sepa— rating another set has become quite widely used. We shall Show that for closed subsets of En, the concepts agree. We 104 first define the necessary terminology. Definition: Let X be a topological space. The subset A C'X pppp X if X-A is not a semi-continuum. A separates X if X-A is not connected. If A cuts X, we may express X-A as a disjoint union of semi-continua which do not meet A. (The term used in the older literature for these semi—continua is "regions com- posants.") If A separates X, X-A is the usual disjoint union of components,ennlif A is closed, recall that these sets are called the complementary domains of A (see p. 8 ). Definition: A pppp (respectively separates) X between p, q if p, q lie in different semi-continua (respectively com- ponents) of X-A. It is clear that if A separates X, then it cuts X. To see that the converse is not necessarily true unless more is assumed about X, A, consider the space X = {(XJH y = sin T/x, Q(xél] U [(O,y)] IYIél] and take A = {(0,0)}. Then A cuts X between (0,1) and (0,-l), but A certainly does not separate X. However, we do have the following Lgmma 8.1: If A C En is closed, then A separates En iff A cuts En. 2322f: If A separates, then it cuts. If A does not separate En, then En-A is open and connected. By [28, p. 116], En-A is path connected, which in our terminology means there is a continuum disjoint from A joining any pair of points in En-A. Therefore, A does not cut En. 105 Thus, we may use the phrases "A cuts B" and WAseparates B" interchangeably, whenever A is closed and B is a con— nected open subset of En. Lemma 8.2: Let A be a closed set in En. Then: (a) each complementary domain of A is a path connected open set; (b) the boundary of each complimentary domain of A is contained in A; (c) if A separates En, but no proper closed subset does, then the boundary of each complementary domain of A is exactly A; (d) if A is compact, then A has exactly one unbounded complementary domain. ,Ppggf: See [28, p. 356], and note that Dugundji's hypothesis of compactness is not required for a - c. Because the separators in (c) are quite important, they are given a special name below. Definition: If a closed set A cuts En between a, b and if no proper closed subset does, then A is said to cut En irreducibly between a, b. If A cuts En irreducibly bet- ween all points that it cuts between, then A is a completely irreducible cut of En. Using a Zorn's lemma argument, it is easy to see that any closed set which cuts E2 between p, q contains a closed subset which cuts E2 irreducibly between p, q. However, if such a subset is to be completely irreducible, then the original cut must determine only finitely many domains [70, 106 p. 135]. Moreover, a cut is itself completely irreducible iff it is the common boundary to all its complementary domains (see the Corollary to Theorem 8.6). The following is due to Mazurkiewicz. Lemma 8.3: Let R be a domain and let S be a complementary Z-H. Then Fr(S) is an irreducible cut of E2 domain of E between all points a, b, where as R, be S [90, p. 193]. 2322f: It is clear that the boundary cuts E2 between such points a. b, for otherwise we could disconnect any continuum joining a,b. Fr(S) C Fr(R) holds, for by Lemma 8.2 (c), Fr(S) C H, and always Fr(S) C S C E§:H. Then Fr(S) C HllE§:H = Fr(R). Let A be any closed proper subset of Fr(S), and choose PGFI‘(S)-A C Fr(R). There must be points xeR, yes in the neighborhood B(p,[d(p,A)]/2). By Lemma 8.2(a), we can join a to x by a path Pl lying in S (and hence disjoint from A), and we can join b to y by a path P2 lying in R. The line Segment S joining a and y lies in the ball, so it too is diSJOint from A. PlUSUP2 is therefore a continuum joining a and b which does not meet A. Thus, A does not separate. We shall also have need of the following result, first Proved by L. E. J. Brouwer in 1910. Theorem 8.4: (Phragmen-Brouwer property) If the boundary of a complementary domain of a continuum is compact, then it is a continuum. Proof: See [102, p. 176], or [ 127, p. 106]. For the more general case in which the boundary is not 107 assumed to be compact, see Wilder [130, p. 48]. Theorem 8.5: If the compact set C cuts E2 irreducibly bet- ween a and b, then it is connected [70, p. 154]. 3329:: Let R be the complementary domain of C con- taining a. Let S be the complementary domain of H C RlJC (by Lemma 8.2 (b)) which contains b. By Lemma 8.3, Fr(S) C Fr(R), and by Lemma 8.2(b), the latter is in C. By Lemma 8.3, Fr(S) is an irreducible cut, as is C. Therefore, 0 = Fr(S). By Theorem 8.4, Fr(S) is connected; hence C is connected. Theorem 8.6: In order that a closed set C should be the common boundary of two domains D1, D2 in E2, it is necessary . 2 and sufficient that C be an irreduClble cut of E between a, b for all aéD bC-D2 [70, p. 133]- 1! Proof: If C is an irreducible out, then by Lemma 8.2 (C), it is the common boundary of D1, D2. 2 Suppose 0 = Fr(Dl) = Fr(D2). 0 outs E between all Points of D1 and D2, since any connected set containing Such a pair of points must meet the boundary, C. We shall now show its irreducibility. . 2 — . . Let D be the complementary domain of E -Dl containing 3 b. Consequently, t)€IB[lD2, and we shall show D3 = D2. From Lemma 8.3, Fr(D5) c Fr(Dl) = Fr(D2). Since all points 0f D2 can be joined to b by a continuum (path) not meeting Fr(D2), the same can be done while missing Fr(D3). Since D2 is connected and D20D3 C C, we have D2 C D5. ' ' domain 0n the other hand, Since D3 is a complementary 108 2 _ _ of E -Dl, D3‘1Dl = C. Thus, Dsler(D1) = C, and hence Daler(D2) = C. Therefore, all points of D3 can be joined to b by a continuum disjoint from Fr(D2) (Lemma 8.2(a)). Hence D3 C D2. D2 = D3 implies Fr(D2) = Fr(DB) = C, so by Lemma 8.3, C cuts E2 irreducibly between a and b. As a corollary, note that C is a completely irreducible 2 iff it is the common boundary of all its comple- cut of E mentary domains. We shall also have need of Theorem 8.7: (Janiszewski) (a) If A and B are continua such that AllB is discon- nected, then AlJB separates the plane. (b) If A and B are compact sets, neither of which cuts E2 between p, q and if AllB is a continuum (perhaps empty), then AlJB does not cut E2 between p, q. 2332;: See [49, p. 192], or [102, pp 175 and 173]. We note in passing that Kuratowski has shown that (a) and (b) are equivalent in any locally connected continuum [75, p. 511]. Lemma 8.8: Let K be a subcontinuum of a compact set C which cuts E2 irreducibly between p. Q. [70, p. 136]. Proof: The result is clear if K = C or K = C. So Then C-K is connected suppose C C K C C. If C-K is disconnected, then by Theorems 8-5 and 4.2, C-K = MlJN, where M, N are nonempty, disjoint, closed subsets of C-K. Moreover, KlJM and KLJN are proper subcontinua of C. By the irreducibility of C, neither cuts 109 E2 between p, q. So by Theorem 8.7 (b), (KlJM) U (KLJN) = C does not out either. This is a contradiction. Therefore, C-K is connected. Lemma 8.9: If a decomposable continuum C is the common boundary of two domains in E2. then C = AlJB, where A, B are proper subcontinua such that C3K = B. and C:B = A. 2322f: By Theorem 4.4, C contains a proper subcon- tinuum K that is not nowhere dense: C C CZK C C. Let A = C:K, B = C:H. Then 63B = 5:55X = C-C-C:K, which is 63K = A by Lemma 5.15. By Theorem 8.6, C is an irreducible cut of E2 between all pairs of points a, b, for a and b in different comple- mentary domains of C. Therefore, by Lemma 8.8, A is a con- tinuum and hence so is B. It follows from the choice of K that A is a proper subCOntinuum of C. A C C imp1ies B C C, and A C C implies B C C. Finally, C = AlJB, for AlJB = C:KlJC:H = (5:237IXEZFEX7 = 5:FFETK7 = C: We have now established the foundation needed to prove both of Kuratowski's common boundary theorems. To continue with the proof of the first such theorem, we prove Lemma 8.10: If C is a compact set which is a completely irreducible cut of E2, and if K is a non-degenerate proper subcontinuum of C that is not nowhere dense in C, then Kflfi-X is disconnected and C—K is a continuum irreducible between all pairs of points belonging to different com- llO ponents of KllC:X [70. p. 136]. 3322;: By Lemma 8.8, C:K is connected. Since C:K C C, c C K, and K UE'K = c, it follows from Theorem 8.7 (b) that KIIC:K is disconnected. Let a, b be in different components of this set, and suppose that L is a continuum containing a, b and contained in m. If L nK is connected, then {a,b} c K n L c K nc—IK' implies that a, b are in the same component. Thus, LflK must be disconnected, whence Theorem 8.7 (a) implies KlJL cuts E2. By the irreducibility of 0, c = K UL. Then Cl‘KcL so that L = C:K. Therefore. C:X is an irreducible continuum. Lemma 8.11: (Straszewicz) If A, B are compact sets which do 2 not cut E and if AlJB cuts E2 into more than two domains, then ArlB contains at least three components. 2329:: See [112] or [76, p. 551]. For the case where AlJB cuts E2 into more than n domains, see [113, pp. 159- 187]. Straszewicz showed in the latter paper that if the union of two continua, neither of which cuts the plane, and which have nflpl components in their intersection, then the union cuts the plane into n domains. He also showed that "n" can be replaced by "countably infinite" [113, 174]. Theorem 8.12: (Kuratowski's first common boundary theorem) . 2 A compact set C which is a completely irreduClble cut of E and which determines at least three domains is either an indecomposable continuum or else the union of two indecom- posable continua [70. p.138]. Proof: By Theorem 8.5, C is a continuum. Suppose it 111 is decomposable. By Lemma 8.9, C = LlJM, where L, M are proper subcontinua of C such that C:L = M and C:M = L. L is not nowhere dense in C. For if it were, then C = CZL = M implies C = S-—M = L, contradicting Lemma 8.9. Ln'cTL = mm = MncTM. We apply Lemma 8.10 to both L and M to conclude that LrlM is not connected and that L and M are continua irreducible between all pairs of points belonging to different components of LflM. By the irreduc- ibility of C, L and M do not cut E2, so by Lemma 8.11, LflM must contain at least three components. By the first part of this proof, L and M are each irreducible continua between all pairs of points in different components of LflM. There— fore, by Theorem 4.11, L and M are indecomposable continua. Eilenberg established a similar result for the sphere 82 in [50, p.82]. We now prove Kuratowski's second common boundary theo- rem. The distinction between Theorems 8.12 and 8.13 is that in the former, C is required to be the common boundary 0f Ell its complementary domains, while in the latter, 0 is only assumed to be the common boundary of ggpg (3) of them. Theorem 8.13: If the plane continuum C is the boundary of at least three domains, then it is either indecomposable or the union of two indecomposable continua [72, p.36]. 7 Proof: We follow the original proof, with only slight modifications. Let D1,D2, D3 be the domains of the hypothesis. C = Fr(Dk), k = l, 2, 3. If C is decomposable, then by Lemma 8.9, there are proper subcontinua K1 and L1 112 in C such that C:X1 = L and C:L = Kl. Suppose further that C is not the union of two indecom- posable continua. Then without loss of generality. L is decomposable: L = K2lJK3, where K2, K3 are proper subcon- tinua of L. Now KllJK2 C C, since if equality held, then C-Kl C K This would imply that L = C-Kl C K2, contra- 2. dieting the decomposition of L. Likewise, KlUK3 C C. Thus C = KllJKzlJKB, where the union of any two of the Ki's is a proper subcontinuum of C. Consequently, there are points xi (Ki, 1 = l, 2, 3 that are not in the other Kj's. Let 6; denote one half the distance from xi to the union of the other two Kj's. Then B1 = B(xi, 6‘) are pairwise disjoint sets such that BinKi C C,3and BinKj = C, for j C i. Since U Ki = C = Fr(Dk), k = l, 2, 3, we have Birle C C, for i = l, 2, 3, and k = l, 2, 3. Because connected open sets in E2 are polygonally arc-wise connected [44, p.108], there are polygonal lines P1, P2 such that Pk C Dk, k = l, 2 and Pklei C C, for k = l, 2, and i = 1, 2. 3. The sets Pklei, k = l, 2, i = l, 2, 3 are compact. So there exist points yie PlrlBi, and zif P2r1Bi, i = l, 2, 3 such that d(yi’zi) is the minimum with respect to the distances between all pairs of points, one from Pllei and one from P2 nBi. Let Ti be a triangle in Bi having yi and zi for the vertices of two of its acute angleS- P1 “Ti = yi. and 113 P nT. = z., for i = l, 2, 3. Since yiEIHJ 216 D2, there 2 i l are points of C in the interior of each triangle Ti, and consequently, there are points of D3 in the interior of each Ti' Thus, we can construct a polygonal line Z C D3 such that ZilTi C C for i = l, 2, 3, since D3 is polygonally arc— wise connected. Let W C Z be a polygonal line minimal with respect to meeting each Ti’ Therefore, W must meet two of the Ti's, say T1 and T3 , only at their endpoints r1, r2: wnTl = r1, wnT2 C C, WUT3 = r3. We agree that (yizi) shall denote either the segment joining those points, or else the line formed from the other two sides of Ti’ depending on whether the segment contains points of W. Thus, Wt)(yizi) = ri, i = 1, 3 and Wf](y222) C C. 1) P1n(yizi) = yi’ P2“ (yizi) = Ziv j- : l, 29 30 Since (yizi) C Ti C Bi’ it follows from the con- struction of the Bi's that 2) (yizi) n (yi+lzi+l) = ‘5 = (yizi) n (Ki+lUKi+2)’ Where the indices i+l, i+2 are reduced mod 3. Therefore, 5) (yizp M, C C. Consider S = (ylzl) U (2123) U (szB) U (yayl), where (le3) and (ysyl) denote polygonal lines extracted from P2, P1 respectively. S is a simple closed polygonal path by equations 1 and 2. Moreover, since W was a polygonal line having only its end— 114 points on S, Sle cuts the plane into three domains, L, M, N such that Fr(L) = (ylrl)lJWlJ(r3y3)lJ(y3yl), Fr(M) = (zlrl)lJWlJ(r323)lJ(23zl), and Fr(N) = S. y l y3 L 1‘ r1 3 M Z Z 1 N 3 By 2), (SUW)nK2 = C. Since K2 is connected, it must be in exactly one of the three domains. It can not be in N, _ ' , 't h s because K2LJ[(y2z2) {y2,22\] is connected by 3) i a points in common with W (since W(](yizi) C C, i = l, 2. 3). and it is disjoint from S, which is the boundary of N, while erw = C. Without loss of generality, K2 C L. Starting from y2, let t be the last point of (y2z2) which belongs to K2. Then (tz2) has no points in common with C except t, and since W C D3, 22 eD2, we conclude that Wr](t22) = C. Therefore, P2lJ(tz2) is disjoint from Fr(L). Consequently, P211(t22) C L, since tEK2 C L. and P2 U(tz2) is connected. This is impossible, since (2123) C P2 and (2123) c S imply that S 0P2 C C, while 3 nL = C. Thus, P2 is not in L. Therefore, the assumption that C was not the union of two indecomposable continua leads to a contradic- tion. We next present some of Knaster's examples of common boundaries of plane domains. Let A be the numbers in I that can be written in base five without using the digits "1" and £3 115 "3". We now describe a continuum Bn that is both indecom- posable and the common boundary of n domains. Fix n712, n "" and let Bn = E Eli9 Fk’ where E0, Ek (O<1{O. See Figure 8.1 (a). The solid, dashed, dotted lines represent respectively the sets D0, D1, D2 to be defined on p. 117. Knaster also discovered the continua {On}, each being the common boundary of n domains and the union of two indecomposable continua [60, p. 274]. Cn = BALJBQ', where ' ' n u ' Bn 18 obtained from Bn by replaClng each Fk by Fk. (x--2k—l)2 + y2 = r2, 2k+lsx, ys o, r EA; (x-2k-l)2 + (y+l-[7/2]5‘m)2 = r2, x52k+l, m 'l v (5 r-l/2) (- A, m70. Bn is obtained from Bn by reflecting it through the line y = -1. See Figure 8.1 (b). The solid, l D dashed, and dotted lines represent respectively D', Di, 2, to be defined on page 117. .XHIHHII! upper c /c 116 ll? The verification that the Bn's are indecomposable can be found in Knaster's paper [60, pp. 274-281]. We will not present it due to its length. However, the essential idea is to show that every proper subcontinuum is nowhere dense, and then apply Theorem 4.4. Knaster accomplished this for Bn by showing that every proper subcontinuum must be con- tained in the union of an infinite number of successive simple arcs (compare Knaster's first semi-circle example, p. 24). The first are in the first union is the semi-circle in F0 starting at (0,0) and terminating at (2,0). The next arc in this union is the semi-circle in El starting at (2,0) and terminating at (3,0), and so on. This union is denoted DO, and it is represented by the solid line in Figure 8.1 (a), D; is the corresponding set in B; , with its first are starting at (l,-l). The set Dk’ for 0<1£(h (b) for 11 odd, [x-(4/5)+2 - 5'(n+1)/212 [37+(l/5)--(7/2)5_(m+1)12 = r2. m7(n-l)/2; + (c) for n even, [x-(4/5)+4- 5-(n/2)]2 + [y+<1/5)-(7/2)5'(m”1)12 = r2, m 711/2. (2) The straight line segments are to have slope +1 and are to join to the x-axis not only the extremities of the semi-circles in MA, but also the latter's points of accumulation. Knaster explains the relationship between Bn and CD on one handJand Bno and COO on the other this way: "It was easy to see that in the case of the continua Bn and Cn’ each domain Rk (O<]{¢er) wormed its way into the next ones by means of an infinite number of narrower and narrower blind alleys terminating in dead ends. They were directed forward and could only reach a neighborhood of an analogous blind alley of a preceding domain after having wound through those of all the following [domains]. This is impossible in the case of an infinite number of more and more distant domains. Now one succeeds in restoring the dense disposition of these blind alleys [in the infinite case] (in order that they should have a common boundary) by occassionally directing them directly backward toward those of the pre- ceding domains. This is precisely the case of the con- tinua B0° and Cup where the first blind alley of each odd region is directed, without winding through the following ones, toward its point on the y-axis . [<7/2)5'(n+1)/2 - 1]. Where "n" designates the number of that domain"[60, p.284]. 121 The fact that there exist indecomposable continua that are the common boundaries of several plane domains allows us to give examples where some familiar double integral formulas fail, as Hocking and Young show [44, pp. 143-145]. Suppose we modify the Lakes of Wada construction by taking the island to be 12. We further modify it by digging suf- ficiently long and narrow canals from the ocean and the two internal lakes so that the three resulting domains, D D 19 D3 each have measure 1/10. Then, if fElq we have that 2' 3 3 [iff = 1, while 2:fo f :: 3/10. Moreover, since 0 Di | l is dense and since C = Fr(Di), i = l, 2, 3, we have 3 Z ITS- If = 3[(l/lo)+(7/lo)] £1. l L Before presenting some examples which show Kuratowski's common boundary theorem fails in E3, we shall mention some of the work done on the "prime end theory" and Burgess' generalization of both it and Kuratowski's result. Caratheodory introduced the term "prime end" in his 1912 paper on conformal mappings [22]. A RElEE 229 of the boundary F of a domain D is the set of limit points of a sequence of nested domains determined in D by a chain of transversals tending to zero in length. A transversal is a simple are contained in DlJF joining two points of F. Trans- versals form a ghain if they are pairwise disjoint and the subdomain Dj‘determined in D by the transversal Tj,con- tains the domain Dk determined by Tk’ for all Tk following 122 Tj‘ This version of the definition was given by Marie Charpentier [23, p. 303]. In 1939, Charpentier investigated irreducible cuts of E2 that were sufficiently complicated so that the entire cut was one prime end. She showed [23, p. 306] that if the continuumC1cuts E2 into two domains and if C has a prime end identical to itself, then C must be either an indecom- posable continuum or else the union of two indecomposable continua K, L such that K = 0:3 and L = 0:K. In 1935, Rutt considered the question of when "the set of prime ends of a plane bounded simply connected domain includes one which contains all the boundary points of the domain." [109, p. 265] He established the following results: (1) "In order that the collection of prime ends of the plane bounded simply connected domain D with boundary F should include one containing F, it is sufficient that F be indecomposable." [109 p.268] (2 V "In order that a plane bounded simply connected domain D with boundary F should have a prime end containing F, it is necessary that F should be either indecomposable or the union of two indecom- posable continua." [109, p. 278] In his thesis, C. E. Burgess investigated continua and their complementary domains in E2. using some results of P. M. Swingle on generalized indecomposable continua. He showed (among other things) that the following theorem holds: 123 "Suppose H is a closed set and M is a continuum in B§:H and intersecting E2-H such that if R1, R2, R3 are three domains intersecting M, there exist three complementary domains of MlJH each intersecting each of the domains R1, R2, R3. Then either M is indecomposable or there is only one pair of indecomposable proper subcontinua of M whose union is M." [19, p. 907] Kuratowski's theorem (8.13), Charpentier's theorem, and the second result of Rutt are now special cases of Burgess' theorem [19, p. 908]. To conclude the chapter, we shall consider the situ- ation in E5. Kuratowski's common boundary theorem fails there, a fact he knew when he published his works on E2 [70, Pa 132], [72, p. 36]. In fact, he gave the following example. Let C be a plane continuum that is the boundary of three plane domains. Join each point of C to a point above the plane and to a point below the plane. The resulting continuum is the common boundary of three domains in E3, but it is certainly neither indecomposable nor the union of two indecomposable continua. . . 3 R. L. Wilder showed in 1933 that there ex1sts in E a peano continuum which is the boundary of three domains [129, PP. 275-278]. He constructed this set by generalizing the Lakes of Wada technique to three dimensions. That is, the island becomes a solid ball, the lakes become two (tangent) balls removed from inside the first ball, and the canals become tunnels. His method can easily be generalized to 124 give a peano continuum that is the common boundary of n domains, or even a countably infinite number, although in the latter case, the diameters of the inside balls must tend to zero. In connection with this, P. M. Swingle has shown in 1961 that if Wilder's tunnels of circular cross section are replaced by tunnels of annular cross section, then the resulting closed connected set in En is indecom- posable [114]. There is a familiar name in the background of Wilder's work: Schoenflies. It is Wilder's belief that Schoenflies' methods of investigating the topology of plane domains and their boundaries could be extended to higher dimensions, even though some topologists felt otherwise [129, pp. 273- 274]. His above paper was a step in that direction. Perhaps the most frequently cited example showing that the three dimensional case differs from the two dimensional one is Lubanski's ANR. which is the boundary of three (or more) domains and which can be decomposed into a finite union of AR's whose diameters are arbitrarily small [81]. (See [28, pp. 151-152] for a definition of ANR, AR.) Thus, We can have a "nice" continuum being a common boundary of three or more complementary domains in E5. Lubanski notes that a mathematician named Gruba con- structed the first such example. It is not known if it had Lubanski's decomposition property, since the paper was lost in WW II and never published [81, p. 29]. CHAPTER 9 ACCESSIBILITY OF PLANE INDECOMPOSABLE CONTINUA In this chapter we shall present another characteri- zation of indecomposable continua in the plane. This char- acterization was given by Kuratowski in 1929, based on work done by Mazurkiewicz in the same year. Kuratowski asked whether every plane indecomposable continuum contains a composant which contains no accessible Point. Mazurkiewicz' surprising answer was that "almost all" composants have no accessible points, in the sense that the union of all composants containing accessible points is of first category with respect to the given continuum [92, p. 107]. In a later paper [93], he showed that in a plane indecomposable continuum, the collection of composants which contain more than one accessible point is either finite or countably infinite. Using the first of these results, Kuratowski showed [74] that a plane continuum is indecomposable iff it is nowhere dense and contains a point which is contained in no Proper accessible subcontinuum. Before proving this theorem, we need to mention the ways in which the term "accessible" is being used here. . . 2 During the late 1920's, a point p contained in a set S C E 125 126 was said to be accessible from E2-S if there were a simple are A having p as an endpoint and such that Arls = p. Maz- urkiewicz used this definition in his above-mentioned papers. Note that this definition differs slightly from the one given on p. 83. However, Kuratowski's definition was dif- ferent than both of the above. X C C is accessible iff there is a continuum L such that LilC = X, and L—C £ C. Instead of establishing Mazurkiewicz' first result, we shall prove Kuratowski's generalization of it. . . 2 Theorem 9.1: If C is an indecomposable continuum in E , the union of its proper accessible subcontinua is of first cate- gory with respect to C [74, p. 116]. Proof: Let a be any point of C, and let P(a,C) be its comPosant. Let Gn k be any complementary domain of the set ’ c UTB a, "mu in E2, and let Fn k be its boundary. Let K(y) Y be the component of ye Fn k contained in C-B(a,l/n), and set Qn = :4 73mm), Q = g on. Suppose D is an accessible subcontinuum of C. By Kuratowski's definition of an accessible subset, there exists a continuum L such that LIIC = D, and L-C £ C. Either Df1P(a,C) # C, or else Df1P(a.C) = C. In the former case, D C P(a,C). In the latter case, Lf1P(a,C) = Ln (C nP(a,C)) = DI'IP(a.C) = p. In particular, a $L. Con- sequently, there is an n sufficiently large that LIIBTETI7HT = ¢. say l/n = [d(a,L)]/2. Therefore, L-(ClJBTaTI7HT) = (L-C) n (L-‘T—IT‘IB a, n ) = (L—C) nL = L-C .4 (23. Thus, there is a 127 k such that Ln Gmk .é {23. Moreover, C .4 an c L-Gmk. Therefore, LnFn k ,4 C. Fn k c C W a, n , and, since Lleia,I7n5 = C, we have C # Llan k C LIIC = D. Hence, DnFnJ, :4 ¢. Thus, momma—3175 = o, and so DnB(a,1/n) = ¢. Therefore, D C C-B(a,l/n), and consequently, D C Qn. Thus, the union of all proper accessible subcontinua of C is contained in QlJP(a,C). Theorem 4.11 (a) shows that P(a,C) is first category with respect to C. Lemma 9.2 will show that Q is also first category with respect to C. Therefore, D is first category. Lemma 9.2: Under the hypotheses and notations of Theorem 9.1, Q is first category with respect to C [92, pp. 112-115]. Outline pf 3322;: We shall show that each Qn is nowhere dense in C. Suppose that the indices n, k are fixed, and that c EC-BTETI7E7; d(c,BTE:I7ET ) > 6 7(3. Let Pl be some composant of C, not P(a,C). By Theorem 4.17, E1 = C, so P1(1B(a,l/n) # C. Let x be in this intersection. and let L(X) be the component of x in C-B(c, 6/2). It is a nonempty proper subcontinuum of C, so by Theorem 4.4, it is a con- tinuum of condensation of C. That is, it is nowhere dense in C. It follows that (PlnB(a,l/n))—L(x) yé (25. Let y be in this set. Since x,y are in P1’ there is a continuum J C Pl C C, irreducible between x, y. (The irreducibility follows from the fact that any continuum containing a pair Of points contains a continuum irreducible between them.) = J-B(a,l/n), J2 = JrlBTETI7n7, and let cl be Set Jl 128 in the nonempty set J11B(c, 6/2). J1 and J2 are closed and satisfy x, ye J2, clé Jl-JZ' and JlUJ2 = J. Thus, the hypotheses of the following lemma are satisfied. Lemma: Let C be a continuum irreducible between p, q of a compact metric space. Let F1 and F2 be two closed subsets such that p, q 6F Fl-F2 C C, and FllJFZ = C. If zéIFl-FZ, 2, and if €‘> 0, then there exists a continuum K C Fl such that KnF2 C C and non-connected, and d(z,K)<.6 [92,pp. 107-109]. Therefore, there exists a continuum M such that M C J1 C (C-B(a,l/n))f1Pl, d(cl,M)1)nV(Pl ) c PlriPl = C. or 1 1 . now apply the following lemma of Mazurkiewicz. 129 Igmmaz Let A be a continuum in E2 which is locally connected. Let f he an uncountable collection of disjoint continua, each of which cuts A. Then if contains three continua such that one cuts A between the other two. That is, two con- tinua are in different complementary domains of the one continuum [92, pp. 109-110]. II I 1" Thus, there are three composants P , P1 , Pl , such I! III that V(Pl) cuts E2-B(a,l/n) between V(P1 ) and V(l>l ). We II '3' claim that either v(P1 )len'k = C, or v(Pl )len,k = C. II II! If not, then choose vl 5V(Pl )nFn’k and vzeV(Pl )Ian’k. V(Pi) cuts E2-B(a,l/n) between v1 and v2 [92, p. 109]. Since v1, v2 €Fn,k’ all neighborhoods of v1, v2 must contain points of Gn k' Hence, V(Pi) cuts E2-B(a,l/n) between some 9 I . . pair of points vi, v2 of Gn,k‘ However, Since Gn,k is a domain, there is a continuum N C Gn,k containing those 2 points. Then N c on k c EZ-(CtJBTEII7ET) c (E -B(a,l/n))- 7 ' 2 V(Pi). This contradicts the fact that V(Pl) cuts E -B(a,1/m between v1 and v2. Therefore, take V to be the one of ' I n I V(P1). V(Pi') which is dlSJOlnt from Fn,k. Consequently, V n g F K(y) = C. e ".k ' But, d(c,V)< e , since d(c,V(Pl)) < 6, so that we have d(C.C-U K(y))< 6 . Since G is an arbitrary point of the 7e; 0 set C-Bia,l7n5 and e > 0 is arbitrarily small, we have that C-U KZyS D C-BZa,I7n5, and CLCE' K(y) D Cf1B(a,l/n). r at 6a,. . 130 Moreover, C = C:Ef§?I7§I U C7TBT§TI7HT. If not, then C would contain either a point of dimension zero, or else an arc of Fr[B(a,l/n)] which would not be a continuum of con- densation of C. This violates Theorem 4.4. Therefore, C-Cénm y = C. Consequently, for any n, k the closed set U K(y) is nowhere dense with respect to C. II,” Thus Q is first category. Theorem 9.3: A plane continuum C is indecomposable iff it is nowhere dense and contains a point which is contained in no proper accessible subcontinuum [74, p. 116]. 2322:: Suppose C is decomposable, say C = AlJB, where A, B are proper subcontinua of C. If C is not nowhere dense, then there is nothing to prove. If C is nowhere dense, then C = Fr(C). Since the set of points accessible from E2-C is dense on Fr(C), both A and B must contain points that are accessible from EZ-C. (To prove the state- ment about density, let p)éFr(C), and let (‘70 be given. There is a point q in E2-C within. 6 of p. Starting from q, let r be the first point of the segment qp in Fr(C). Then qr lies in EZ—C, except for r, and hence r is acces- Sible from EZ-C and is within. 6 of p.) Let p, q be accessible points of C contained in A, B respectively. Thus, there exist simple arcs L, K having P. q as respective extremities, and such that Lth = p, KIIC = q. Then LlJA, KlJB show that A, B are accessible in the sense of Kuratowski, Since the union of these proper Subcontinua contain all points of C, we have contradicted 131 the condition given in the theorem. If C is indecomposable, then C is nowhere dense in E2 by an easy corollary of Theorem 4.4. The accessibility condition follows by Theorems 9.1 and 4.8, the latter of which says that no T2 indecomposable continuum is first category with respect to itself. CHAPTER 10 TOPOLOGICAL GROUPS AND INVERSE LIMITS A. D. Wallace nicely expressed the reaction of many mathematicians to the concept of indecomposable continua when he said, "We commonly think of indecomposable spaces as being monstrous things created by set—theoretic topol- ogists for some evil (but purely mathematical) purpose." [123, p. 96] We hope to dispel any such feelings about indecomposable continua by showing that they can play a role in areas other than point set topology. In particular, we are going to explore two roles of indecomposable continua in topological groups. First, we shall present Wallace's proof of the fact that if we have a continuous multiplication with a two sided identity defined on a continuum, (such a structure is called a plan) then the clan is a topological group, provided the continuum is indecomposable. The second part of the chapter will be devoted to the famous class of examples called solenoids, which are both indecomposable continua and topological groups. We begin by defining some of the above terminology. A 33922 is a set G together with an operation, *, such that: (a) a,bGG implies a*b (G; 132 133 (b) * is associative; (0) there is an identity element e, such that x*e = x = e*x, for all x:6G; (d) for each x:sG, there is an inverse x‘l, such that xx.1 = e = x-lx. If only a and b hold, we have a semi-group, while if only a, b, and 0 hold, we have a monoid. If all of the axioms hold and the operation is also commutative, the group is abelian. Definition: A topological group is a set G which has both a group structure, (G,*) and a topological structure (G,T) such that: (a) f: GxG-—9G given by f((x,y)) = x*y is continuous; (b) g: GJ-9G given by g(x) = x-1 is continuous. For example, the real numbers under the usual addition and topology form a topological group. Later in this chap- ter we shall need the fact that (z! z is complex and (zl = l} is a topological group under complex multiplication. Its underlying topological space is 81. A mgb is a T2 space (S,T) together with a continuous associative multiplication, m. If (S,T) is also a continuum and if (S,m) has a two sided identity, then the mob is a _ci_ap [123, p. 96]. Let C C L c S, where s is a mob. L is a lgfi $922; if SL C L. Similarly, one defines a Eight M and a H2 and £1821. The next theorem shows that the algebraic property of existence of inverses in a clan is implied by the topo— 134 logical property of indecomposability. Wallace first proved the result in 1953 under the additional restriction that the space be metrizable [122]. He later simplified it [66], but the version we give appeared in [123, p. 103]. Theorem 10.1: An indecomposable clan is a topological group. 2322f: Let S be an indecomposable clan. Let H denote the maximal subgroup of S containing the identity. Let K denote the minimal closed ideal of S. To prove its exist- ence. let {1030‘ E ,7 denote the set of all closed ideals of S. The collection is nonempty since S is in it, as well as the closure of any ideal. If L and R are respectively left and right ideals, then LflR C C, for x EL and ye R show yKZELflR. Thus, [Id }d 6.7 is a collection of closed sub- sets satisfying the finite intersection property. Since S is compact, K : n IX where D is a directed set. We will use the notation {x 0/qu é .7 to rep— resent the range of W . A net {XNBM e 07 converges to xéX if for all neighborhoods U(x), there is an a (D such that for all b wa, xb 6U. A net {x,dd 6'7 accumulates at x (X if for all neighborhoods U(x) and for all a 5D, there is a bED, b7/a, such that xb 6U. W : F->X is a subnet of T: D—>X iff there exists a function f : F -)D such that Y = cPof and for each In 6D, there is n in F such that if P7/n, then f(p) 7,m. Lemma 10.2: Consider (G,m,T), where (G,m) is a group and (G,T) is a compact T2 space. If m is continuous, then so iS g, where g(x) = x-l. That is, (G,m,T) is a topological group, _P_rgp_f: Let x6 G be arbitrary, and let (x(}_‘ e' 07 be any net converging to x. To show that g is continuous, we must show {g(xx )}fig(x) [58, Theorem 3.1 (f), p. 86]. {g(X.‘ )} is a net and since S is compact, there is a subnet {g(xq )} converging to some unique yé S [58, p. 136]. (1 137 {x40} converges to x since {xx} does. {(x"! , g(x.(p ))} is a net in GxG and it converges to (x,y). Since 111 is continuous, {m(xxp ,g(xup )} = {e} converges to m(x,y) = Xy. Therefore, xy = e, which implies y = x'1 = g(x). Hence, {g(xocp )} Cag(x). To see that {g(x°,)} converges to g(x), we note that if {g(x,([} has any other accumulation point z, then the above argument can be applied again to get z = x-l, so that Z = y. Hence, g(x) is the unique accumulation point of {g(xo. 15'. Since S is compact, {g(x_,%must in fact converge to g(x). For otherwise, there exists a neighborhood U[g(x)] such that for all a ED, there exists b 6D, b7/a, such that g(xb)¢U. Then choosing aof D, there is a17,aO such that g(xal)¢U. In general, given a, such that g(xax )¢U, there exists a@ 7, a). such that g(xa0)¢'U. Then{g(xad)} is a net none of whose terms is in U. Hence. it can not accumulate at g(x). Since S is compact, it accumulates elsewhere, contradicting the uniqueness of accumulation point. There is a great deal of information in the literature on the subject of when an algebraic structure and a topo- lOgical structure are sufficiently compatible to yield a tOpological group. See,for example, the references at the end of Wallace's paper [123; llOvll2]. Also, the book by Husain [46] does a great deal in this direction. 138 It is interesting to note that the conclusion of Theo- rem 10.1 holds if the hypothesis of being indecomposable is replaced by that of being a manifold [122, p.2]. This is especially curious since, in Wallace's words,"certainly manifolds and indecomposable continua are antipodal points on the sphere of topology." [122, p. 2] The question naturally arises as to whether there are any indecomposable continua that are also topological groups. We would certainly hope so, for otherwise, the preceding theorem would be of very little interest. In fact we shall show that all solenoids have the desired properties. Since one of the most useful ways of describing these spaces involves inverse limits, the next few paragraphs are devoted to stating some properties of inverse limits. Paul Alexandroff introduced the concept in 1929, although in a slightly different form and context than ours [2]. The following material follows the treatment in Eilen- berg and Steenrod [31], Chapter 8, except where otherwise noted. Let M be a directed set. A subset M' C M is ggfipgl in M if for each méM, there is an m' 6 M' such that mtm'. Let X = {X¢;}q e/W be a collection of sets and F = {f(p]- a collection of functions such that whenever a 5 fl , then f..@ (Xp)cx,,,, for all oxen, f...,(x,.) =X.,, and o - 9‘1. I such a case X fuefpV-foar’“-‘p n ’{“’ + f is called an inverse system Q: sets. If M = Z ’ «’0 ‘M ._______ ____ 139 we say it is an inverse sequence. The sets X... are called the factor spaces of the system, and the functions F are called the binding m. If each Xat is a topological space and each binding map is continuous, then we have an inverse system of topological spaces. If each X“ is a topological group and each map is a continuous homomorphism, then we have an inverse system of topological groups. The inverse Emit of an inverse system of topological spaces (or groups) {X,, ,f.,(,} is LiJ (X ) = Xco = {x = (x...) 6 71X¢\u5(3 =) ffipopfl (x) = p~(x)}: where 77X ,( is the product of the spaces X a. (see [28,pp. 21, 98]) and p3,:71X‘ -—>X2§ is given by P7! ({xu}) = xx , X00 is given the topology it inherits as a subspace of 71X,” If each X.‘ is a topological group, then X” is a topo- logical group. Lemma 10.3: If {XOU fag} is an inverse system of topo- logical spaces (groups), ' then each p a is a continuous function (homomorphism). 3322;: See [31, p. 216]. Lemma 10.4: Let {X,,. , fa, a} be an inverse system of topo- logical spaces. (a) If for each 0:5 (3 , f“ p is one—to- one, then p.‘ is one-to-one; if fuf’ is one-to-one and 011130. then so is pa . (b) If the index set M is countable or if each X o. is a compact T2 space, and if each fN p is onto, then pm is onto. 3392;: See [31, pp. 216, 218]. Lemma 105; Let [Xv , fag] be an inverse system of topo- 140 logical spaces. (a) If each X x is T2, then Xe, is a closed subspace of WX, . (b) If each X o. is compact and T then 2, X” is compact and T2. If also each X,x C C, then Xe, C C. 2339;: See [31. pp. 216-217]. Lemma 10.6: If each X.‘ in the inverse system is a nonempty T2 continuum, then X00 is a nonempty continuum. _Pr_oo_f: See Engelking [32, p. 244]. Lemma 10.7: If {Xuquc} is an inverse system of spaces, -1 then a, (U), where U runs over all open sets of X, and °< over M, is a basis for the topology of X00. Proof: See [31, p. 218]. I I } If we have two inverse systems {Xu ,fup} , {Xx ,fdp over M. we can sometimes define a collection of functions I I {TOD}: {X«,fqp} “fig“,fpp} , (f2: X“——>X« such that if 0( g (3 then the following diagram commutes: X {r1 X a fix? P 7.. (Fe x15? . X (3 «e In such a case, {19“} is called a map from {X,(,f« (3} to {X'oc ’f'd a} . ' I 0 Lemma 10.8: Let {Xa ,fqp} , {Xat ,fm g} be inverse systems of topological spaces (groups) over M with a map { Tu}. Where each of the ch's is a continuous function (homo- 141 morphism). Then {‘?«} induces a continuous function (homo- mor hism : ' ' p ) CF” Xm—9XOD given by fi°~({x~}) s {find}, Proof: See [31, p. 218]. I O l I Lemma 10.9. Let {Cfix}. {xwfw} -.[X,, ,fue} be a map of the systems. If each ‘Rx is one-to-one, then so is the induced map $90,. If each (qup“ is onto, then Cam”) is dense in X . He ' ' ' an nce, if each X0‘ is compact, qt, is onto. Proof: See [28, p. 430]. Lemma 10.10: Let M be a directed set, M' C M cofinal. Let [Xohfoqfi be an inverse system over M and {Xd'ifu'p'} the inverse system extracted from the first by choosing each u', ' I F to be in M . Then Lim Xg, is homeomorphic t0 Lim Xof. e— e—* Proof: See [28, p. 431]. As our final result of this section, we present a recent (1971) theorem of Kuykendall giving necessary and sufficient conditions for an inverse limit of an inverse sequence of metric continua to be indecomposable. We prove only the sufficiency, since that is the only part that we will be using in the chapter. Th - ' eorem 10.11. Let {Xn' fn,m} be an inverse sequence such that for each n, Xn is a non-degenerate metric continuum with metric d and surjective binding maps. The following n’ are equivalent: (a) X00 is indecomposable. (b) If n is a positive integer and 67’0, there is a positive integer nr7n.and three points of Xm such that if K is a subcontinuum 0f Xm containing two 142 of them, then dn(x’fn,m[K])<'e for each x eXn. 2322:: For (a) implies (b), see Kuykendall's thesis [77]. We only note that this proof makes use of Theorem 4.11, as we would expect. (b) implies (a): Suppose that (a) does not hold. Since M is non—degenerate, there are proper subcontinua H and K of M such that M = HlJK. There is an n such that pn(H) and pn(K) are proper subcontinua of Xn: Since H is proper, there is an x = {xi} 6X” -H. For each y = {yi} 6 H, there exists iy such that x. C yi . Further, if n7,iy, then y y Xn C yn. Otherwise, we would have xi = piy(x) = fiy,nq%£x) = fi ,nopn(y) = piy(y) = yiy, a contradiction. Thus, for each y: H, there exists an open set (see -1 Lemma 10.7) Ui = pi (Xi -{xi }) such that pi (x)npi (Ui ) y y y y y y y Since H is compact, there exists a finite subcover of =,¢Uiy{ 3K . Then, for n17,max{iyj}, xnl C ynl , for all ‘ h t H _ {yi} 5H. Hence pnl(x)¢1%h(H), and we have t a pnl( ) is a proper subcontinuum of Xn . Likewise, there exists n2 1 Such that pn (K) is a proper subcontinuum of Xn2' Setting 2 n = maxfnl,n2} gives the desired result. Thus, there is a point qleXn-pnm) and an 6, 7 0 such . . _ d that dn(q1,pn(H))), 5.. There is a pOlnt qzexn pn(K) an 143 an 5,) 0 such that dn(q2,pn(K)) 7/ 62. Let 6 = min{€.,6,}. Iflnwrgand a, b, c are three points of Xm, then two of a, b, c are in one of pm(H), pm(K). Since fn,m’pm(H) = pn(H), fn’m°pm(K) = pn(K)7 dn(ql’pn(H))7’ 6 y and dn(q2,Pn(K))>/ 6 , it follows that (b) does not hold. We can now define and study the solenoids. Let {aj}be a sequence of real numbers with iaj’7 1. Let S1 = {z' z is l l . complex and [z] = 1}. Let fn,n+l: S ————9Sn be given by an n n+1(z) = z , n = l, 2, . . . . The inverse limit of f this sequence is called a solenoid, 2:. If the sequence {ad} is replaced by a sequence P = {p1, p2, . . .}, where the pj are prime (1 is not considered to be a prime), then the resulting inverse limit is the P-adic solenoid, 2::p. If all the pj's are equal to say p, then we have the p—adic solenoid, :E:p. However, if we took p = 1, then the circle results, and we say that the solenoid is degenerate. Historically, 2:: was the first solenoid to be discovered, 2 although its original formulation will be given later. We make one last preliminary remark about 2::pu Note that the set of all P-adic solenoids contains the set of all sole- noids whose binding maps are zn, where n is any integer and may vary with the binding map. For we can replace this binding map by a finite sequence of factor spaces and binding maps which arise from the prime factorization of n. We now discuss some properties of 2:: . Note that since each factor space is a non-degenerate T2 continuum, 144 Z is a non—degenerate T2 continuum. Moreover, since each factor space is a topological group and each binding map is a continuous homomorphism, Z: is a topological group. The indecomposability of 2: follows at once from Theorem 1 10.11. First, any subcontinuum K of Sm is an arc of the form {claim (- 9 £- 6, 04 F-‘X C l T}. Then, given n 7,1, - '7"; ‘0“. C- > 0, choose the three points to be e10, e'1 , e4 in l . Sm’ where myn is large enough that [am-1 . . . . - n17’4’ which is possible since lajl )1. Then, if K is any subcon— tinuum of Si containing any two of the above points, K must contain an arc of length m/2 containing them. The image of this are under fn, has length lam-1 . . . . ' an“?! /2)7/2 77. m Hence, fn,m(K) = Si, so that for any x6831, d(x,fn.m]K]) = 0 < E. The indecomposability of Zp can also be shown directly. Suppose Zp = A UB, where A, B are proper sub- continua. There exists n such that pn(A) C Si C pn(B), (see p. 142). Since pn is continuous and A is a continuum, pn(A) is a continuum. Hence, Six—prim) must be an arc sans end— points, say K = {819 ] N L 9< (3 , P-°< 52 rn'}. Then we have _ Q's - ”n17, 177 =fn];n+1(K)=EJ {eff %\+ 7 2 Y4 1%. +117} -1 1 = U I’m' Moreover, pn+l(A) C fn.n+l[pn(A)] C Sn+l-L' But L is the disjoint union of the qn arcs L1, L2, . . ,,an, and since p (A) is connected. we must have pn+l(A) c J I n+1 145 where J is one of the components of Si+l-L. Thus, the arc length of pn+1(A) is strictly less than 77. Likewise, the arc length of pn+l(B) is strictly less than 77. Therefore, the arc length of pn+l( 2 p) = pn+l(AlJB) = pn+l(A)U pn+l(B)< 2 7r. contradicting the fact that the projections are surjective (Lemma 10.4 (b)). This is essentially the way the indecomposability of 2:. was proved by A. van Heemert in 1938 [119, p. 323], although he did not supply the details of the arc length argument. D. van Dantzig described each :E::g as an intersection of solid tori in 1930 [118, pp. 102-125], but he did not mention indecomposability. We shall give this description later and show its equivalence to the inverse limit defi- nition. H. Freudenthal announced the indecomposability of :E::n’ described in terms of inverse limits, but he gave no proof [40, pp. 232—233]. Finally, Vietoris, the dis— coverer of the first solenoid, 2::2, mentioned it was indecomposable, but again he gave no proof [121, pp. 459— 460]. This is not surprising in view of his formulation of 2::2, which we now present without further comment. Let F denote the Cantor set in the unit interval,I, and consider FxI. For each x6 F, identify (x,0) with (f(x),1), where f is defined on F as on p. 146. For con- venience, we use the triadic expansion for the numbers used in defining f. Let x (F? 146 x+0.2 if 0£X£O.l X + 0.12 -1 if 0.2 5x:£0.21 x + 0.012 -1 if 0.22 éx:g0.221 f(x) = x + 0.0012 -1 if 0.222 ex:§O.222l 0 if x = 1 See Figure 10.1, p. 147. For a discussion of why this is homeomorphic to the toroidal description, see [118, pp.106- 108]. The solenoids 2:}nare often described as the inter- : section of solid tori. The basic procedure is to put one } torus inside another in a special fashion. Namely, the torus Tn+1 must be wrapped longitudally pn times around the inside of Tn' Thus, for p1 = 2, T2 is embedded in T1 as shown: I ‘ 147 (b) (a) (C) Figure 10.1 148 More precisely, let T denote the solid torus, T = SlxD2 =1{(s,z)][s|= 1, \zlé l, s. z complex}. Define g from n,n+l P Tn+1 to Tn by gn,n+l((s,z)) = (s n, [z/cn] + [s/2]), where 0can be given as w 0 g1 n(T . To see that this is homeomorphic to the "unit 9 n) circle" definition, we construct another inverse sequence. Let Xn = gl,n(Tn)’ n = l, 2, . . . , let h = i, the n,n+1 inclusion map, and consider the inverse sequence X ,h . n n,n+1 We first show that Lim [Xn’h is homeomorphic to 6.. n,n+1] oo 7 gl,n(Tn)' Choose x = [Xi]é'£iE {Xn'hn,n+I]' Since the binding maps are all inclusion maps, i(xn) = Xn-l = x for all n. n! Thus, all the coordinates of x must be the same, say x0. Hence, x0 is in each factor space, and therefore, we have 09 x050 gl’n(Tn). Define (P: Iéi_m {gl’n(Tn),hn,n+l} —> on (‘1 gl,n(Tn) by [({xJfl = x0. ‘P is clearly one-to-one and onto. Since the domain is compact and the range is T2, we need only show that T’is continuous. But, the map ?: Lgpu{gl,n(Tn), hn,n+1]‘—9ml is just p1, the first pro- jection map, which is known to be continuous (although it is not onto). Therefore, f’: Big {81 n(Tn)’hn n+1] “‘T> 151 w 0 I q>X°°. Moreover, since each g1,n is a homeomorphism, so is G. Thus, 2::p, as first defined, is homeomorphic to fielmwn) (by (race ‘83). An obvious question to ask is this: are the solenoids 2::P homeomorphic? The answer is no, as the following theo- rem shows. First, some terminology. Let P, Q be sequences of prime numbers. P is equivalent to Q if a finite number of terms can be deleted so that every prime number occurs the same number of times (possibly infinitely often) in the deleted sequences. Theorem 10.12: :13 is homeomorphic to ZQ iff P is equivalent to Q. Thus, in the case of Zn and Zm’ in is homeomorphic to 2::m iff m and n have the same prime factors. 239.923 Van Dantzig was the first to prove this for an [118, p. 122]. Since his proof was based on the torus 152 description of a solenoid, it was fairly involved. The first proof based on inverse limits was given by McCord in 1965 and relies heavily on Cech cohomology theory [83,p. 198]. (Cook has also generalized van Dantzig's results, but in a different direction [25].) For the sake of completeness, we include McCord's proof, but lack of space precludes introducing all the terminology needed to make it self—contained. "From the continuity theorem for Cech cohomology [31, p. 261], one sees that Hl( Z:IHZ) is isomorphic to the group FP of P-adic rationals (all rationals of the form k/(plc . . .~pn), where k is an integer and n is a positive integer.) Also it can be seen that 2::p, as a topological group, is topologically isomorphic to the character group of FP' By number-theoretic considerations one can see that FP is isomorphic to F iff P is equivalent to Q." See his thesis Q [82, pp. 22-26] for more details. As our next major result (10.17), we shall give van Heemert's answer to the question of which metrizable compact connected abelian topological groups are indecomposable. Theorem 10.13: Let G be a metrizable compact aonnected abelian topological group. Then G is topologically iso- morphic to the inverse limit of a sequence of Lie groups. In particular, these factor spaces are metrizable compact connected locally connected abelian topological groups, and the binding maps are continuous surjective homomorphisms. Proof: The first statement is proved in Husain [46, 153 p. 154], although its original discoverer seems to be H. Freudenthal [39, p. 69]. In the proof, the n322 factor space was shown to be G/Nn' where {Nn] is a decreasing sequence of closed normal subgroups. Hence, each factor space is abelian. Since the binding maps are continuous and onto and the index set is countable, the projections are continuous surjections. Thus, each factor space is compact and con- nected. Since each factor space is a Lie group, it is also locally connected. The metrizability follows from the con- struction of the factor spaces. Theorem 10.14: Let H be a metrizable compact connected locally connected abelian topological group. Then H is topologically isomorphic to a torus group Tk = fi'Sl, where k is a positive integer or SU;. 2332:: See Pontrjagin [104, p. 380]. Thus, the G of Theorem 10.13 is topologically isomor- phic to @{Tkm fn’m}. It will be useful to express a torus Tm as Em/Zm, where Z denotes the integers. If h: E-—)T is given by h(x) = eix, then Em/Zm is homeomorphic to Tm under the map HIn = 7[h. Moreover, if we have f: Tm——§Tn which is a continuous sur- jective homomorphism, then there exists a continuous sur— jective homomorphism F: Em——) En such that fon = Hno F [14, p. 82]. Such an F is, of course, a linear transformation. Moreover, we see that if we have an inverse sequence of k torii, T n, with surjective binding maps, the dimension, kn, must be a non-decreasing function of n. 154 Lemma 10.15: Let f: En——§Em,11wn1wl, be a continuous sur- jective homomorphism. Then f is monotonic. That is, if C C Em is connected, then f-l(C) is connected. P_roo_f: Let ker (f) = {xennl f(x) = (0,0, . . .,0)_7y. It is well known that ker (f) is a linear subspace, and since f is onto, it has dimension k = n-m. Therefore, ker (f) is topologically isomorphic to Ek. Moreover, En ~E InkxE gr [En/Ek]xEk; let E denote this composite homeomorphism. Since f is onto, it is easy to see that f is an open map. Hence, by [28, p. 130], the fol- lowing diagram commutes: [En/Ek]xEk 5 ;En f 7‘ x En/Ek/ where p is the natural projection, and g is the natural . -l “ -1 homeomorphism. Therefore, f (C) = g?» P [g(C)] = -l 'I §a[g(0)xEk]. Since g(C) and Ek are connected and E is a homeomorphism, f-1(C) is connected. Thus, f'l(C) is homeomorphic to CxKer (f), where C is the obvious homeomor— phic image of C in En. Theorem 10.16: Let {W fn fm] be an inverse sequence of kn-dimensional torii with binding maps that are continuous surjective homomorphisms. If there is an n such that kn7IL then him-{Tm fn m is decomposable [119, p. 322]. Proof: Let m be the first integer such that km27l. By n . . Lemma i0. .10, Lim-{Tk, fn fn+1]n.yl is homeomorphic to Lim {T n, b There exist proper subcontinua Am, B n,m]n7,m' m 155 wh . . km -1 -1 ose union is T and such that H (A ) H (B ) are con- km m ’ km m k nected in E m: km i9 10 T ={(e \ ,0 o 99e km))0é9ié2fnr, i = 1, . . . ,km};1et 19 19 Am={(e lyoooge Kn“)|0 $915277, 1:1,...,km_1, 05 ohms 7T} u '0 ((el' ,. .. ,eigmm){0 = 91’ i = 1,. .. ,k -1, ogm%42wh mm B ={(e19., 000 yelofi’mHO 591‘: 277, i = 1y o o c 9k '11 m 4 . OK.“ i0 10 - {(8 ’700‘96 KM)|O=91,1=1,...,km-l, 05 Omh We next construct an inverse sequence of proper sub- km+n -l B of T . Let Am+l = f 1(Am), continu a Am+n’ m+n m,m+ _ —l . Bm+l _ fm,m+l(Bm)’ These sets are closed in the compact km 1 Space T + and hence are compact. Moreover, we have seen that the following diagram commutes: k me tHkM+l k Tkm if in», ......o T m+l -1 _ '1 '1 A B choice of Therefore, fm,m+l(Am) _ Hkm+lorm,m+1°Hkm( m). y Am and Lemma 10.15, Am+1 is connected. Similarly. Bm+1 is connected, and it is clear that these are two proper sub- 156 - - km+1 continua whose union is T . = f'1 ( Proceding inductively, we define X m+j,m+j+l m+j+l Xm+j)’ where X is A or B. As above, the sets are easily seen to be compact and proper. To show that they are con- nected and cover, consider the following diagram: k . k _ k . E Igj-J-l FM-riv'uvua. 4 E 13:3 Fmrj’m+j+0 *E m+J+1 H , ,, . Kmfi" km” 9 “mu-J +8 wk . ~1k . k . T 9:3-1 4M‘PJ-‘JM#J‘ I; iii-+3 in", j‘m+‘i$I T m+J+l ‘ ' r Since the two small blocks commute, so does the large one. We can establish the desired properties as before, since -1 -1 . -l k . H (X 111+.) H (Xm+j) = Fm+j-l,m+j km+j~1 m+j-l)° LetX=Lim{X f ‘5. m+n, m+n,m+n+i}! X = A, B. we Claim that km+n A and B form a decomposition of Y = £39'[T , m+n,m+n+i]; A and B are subcontinua of Y by Lemma 10.6. AlJB = Y holds, km since given y = [yi] eY, we have ym eT , and hence either in Am or Bm. Without loss of generality, suppose ymcFAm. '1 ‘ ce eY. Inductively, if Then ym+l 6 fm,m+1(ym) C Am+l’ Sin y -1 . There- m+j,m+j+l(ym+j) C A . . ym+j éAmt)" then ym+j+l6f m+j+l fore, y e A. Finally, A, B are proper subcontinua. Choose y={Ym+i} EEAm-Bm, which exists since the projections are surjective. f-l - = B Then ym+1€:fm]m+l(ym) C Am+l' Since Bm+1 m,m+1( m)’ 157 then ym+l ¢ Bm+l‘ Again induction shows that ym+ié AIn+i - B for all i c 2*, so that yé A-B. Likewise, B-A ,4 C. m+i’ Theorem 10.17: Let G be a compact connected metrizable abelian topological group. G is indecomposable iff G is a non-degenerate solenoid 2:.. 2322f: We have only to examine the binding maps from Theorem 10.13. By [14, pp. 9, 82], the binding maps must have th f. . e form j,j+l a . (C) = e j 19, aje E: if they are merely assumed to be into. For them to be onto, we need ’ajl7yl: given any eiv’é S1, it has ein/aj as a pre-image, provided Waj l 2 77. That is, provided v24 27! [a3] . Taking Ly arbitrarily close to 2'W forces ‘33] ml. However, if all but a finite number of the aj's have absolute value 1, the solenoid is degenerate and hence decomposable. We know from their toroidal descriptions, that 2:]; can be considered as subspaces of E3. It is natural to ask if these solenoids can actually be embedded in the plane. The answer is no, as the following stronger result shows. Theorem 10.18: A solenoid 2::p is not a continuous image of any plane continuum. 2322:: Fort first proved this result for the dyadic solenoid [P = (pi), pi = 2] in 1959 [38, p. 512]. In his Yale thesis (1963), McCord established the theorem as stated for a general class of spaces which include ZEIP [82, p. 80]. The embeddability question seems to have been answered earlier, at least in the folklore of the subject. CHAPTER 11 OPERATIONS 0N INDECOMPOSABLE CONTINUA It is natural to ask whether indecomposability is pre- served by any of the usual set-theoretic operations. It is clear that if A, B are two indecomposable continua, neither contained in the other, then, even if the union is connected, it is decomposable. However, Ale may be disconnected, even if it is nonempty, so this case does not appear to be too interesting either. Of course, if we assume the continua are hereditarily indecomposable and the intersection is con- nected, then ArlB is hereditarily indecomposable. There is no hope for products at all: Theorem 11.1: If A and B are non-degenerate T2 continua, then AxB is decomposable [119, p. 319]. 2322:: Let A = ClJD, where C, D are proper compact sub— sets of A. (x, yeA and A being T2 imply there exist U, V open disjoint sets containing x and y respectively. x¢7, yew, since unv = C. Take 0 = t, D = 17?.) Let b€B be arbitrary. We claim that the following is a decomposition: AxB = [(03:13) u (Ax[b})] u [(DxB) u (Ax{b})] :- 01 u02. If C is connected, then Cl is clearly connected. If C is disconnected, and if {Fu}- are its components, then C U F and c = [(u F )xB] u [Ax{b}] = [U(Fq xB] u [Ax{b}] x ' 1 x a K a 158 159 = g [(Foch) U (Ax{b})]. Since Fe, xB, Ax{b} are connected, and since F0. C C C A, b 6B, then (F... xB) U (Ax{b}) is con- nected. Since each set contains the connected set Ax{b}, the union, 01’ is connected. Cl is closed in compact AxB, so Cl is a continuum. Likewise, C2 is a continuum. Both are proper subcon- tinua of AxB. For C C A implies there is a (116 D-C, and B non-degenerate implies there is a bl C b in B. (dl,b1)¢Cl. Likewise, 02 is proper. However, the situation changes for inverse limits. J. H. Reed proved the following in 1967. Theorem 11.2: (a) Let {X,,( ,fup ,07} be an inverse system of T2 indecomposable continua over a directed set 07, where the binding maps are continuous surjections. Then the inverse limit, X00, is an indecomposable continuum. (b) If each of the above X9, are also assumed to be hereditarily indecomposable while the binding maps are only assumed to be continuous, then X00 is hereditarily indecom— posable [105, pp. 597-599]. M: (a) We have seen (Lemma 10.6) that X00 is a continuum. Suppose X00 = A UB, where A, B are proper sub- continua of Xw. As in the proof of Theorem 10.11, there is an o< such that p9, (A), p,x (B) are proper subcontinua of ch' Since p9, is surjective, we have X0. = p,)( (A) Upo, (B). contradicting the indecomposability of X“. (b) Let K be any subcontinuum of X00, let KO. = p0, (K), - . Each K is an indecomposable andletgdp_ “51K“ OL 160 continuum. {K,x ,g u p , ’7} is an inverse system with con- tinuous surjective binding maps, and K“, is homeomorphic to K [21, p. 235, #2.8]. By (a), K is indecomposable. We shall see in the next chapter that, under suitable modifications, a similar theorem holds for pseudo-arcs. We now consider mappings of indecomposable continua. A continuous image of an indecomposable continuum need not be indecomposable, as the projection of Knaster's first semi-circle example onto the unit interval shows. On the other hand, homeomorphisms clearly preserve indecompos- ability. Is there any type of mapping satisfying inter- mediate conditions that preserves indecomposability? The answer is yes, as the following theorem shows. Theorem 11.3: Let X be an indecomposable continuum and let f be continuous and monotone. Then f(X) is an indecompos- able continuum. nggf: Since f is continuous, f(X) is a continuum. If f(X) = AlJB, where A, B are proper subcontinua, then we have X = f-1 (A)lJf-1(B). Since f is continuous, f-1(A), f-1(B) are closed and therefore compact. Since f is monotone, they are connected. A-B C C C B-A implies C C f'1(A-B) = f-1(A)— f‘1(B) and C ,e f’1(B-A) = f'1(B)-f’1(A). Therefore, f'1(A), and f‘l(B) are proper subcontinua , contradicting the indecomposability of X. We conclude this chapter with the following rather startling result of J. W. Rogers Jr.: There is a plane indecomposable continuum that is a continuous image of every 161 indecomposable continuum. Consider the inverse sequence [In’gn,m] where, for each natural number n, In is the unit interval and 2x if 0 5x £l/2 gnbc) = {2-2x if 1/2 exal . Let D denote the inverse limit of this sequence. By Kuyken- dall's theorem (10.11), this continuum is indecomposable (take the three points to be 0, 1/2, 1 and choose m = n+1). Theorem 11.4: Let M be any metric indecomposable continuum. Then there exists a continuous function f such that f(M) = D. Pr_oqf: See [108, p. 452]. It is a "folk theorem" of the subject that D is actually homeomorphic to the first Knaster semi-circle example [108, p. 450]. CHAPTER 12 HEREDITARILY INDECOMPOSABLE CONTINUA In this chapter, we shall survey some of the more important results of the last twenty-five years in the study of indecomposable continua. We shall be dealing with hered— itarily indecomposable continua in general and with such special cases as the pseudo—arc and the pseudo-circle. The change in emphasis of this chapter from ordinary indecom— posability to the more restrictive hereditarily indecompos- able continua reflects the changing areas of major interest in the investigation of indecomposable continua. We shall also see that certain examples of hereditarily indecompos- able continua have been studied intensively because of their relationships to long-standing problems in plane topology. Not only has the subject changed directions since the late 1940's, but it has also undergone a "change of person— nel." That is, most of the work done on indecomposability prior to then was done by Europeans, primarily from the Polish school of mathematics. However, since 1948 most of the work seems to have been done by Americans, primarily by first, second or third generation R. L. Moore students. (For an interesting account of Moore's famous teaching method, see the paper by Lucille Whyburn [128, pp. 35-39].) 162 163 We shall also see that some problems originating in the Polish school were either partially or fully solved in the last quarter century by Moore descendants. We recall from Chapter 6 that Knaster discovered the first example of a hereditarily indecomposable continuum in 1922. His motivation was simply to prove that there exists a continuum each of whose subcontinua is indecomposable. Many of the examples of hereditarily indecomposable continua presented in this chapter were constructed in order to have an example of a continuum satisfying property P, where P was something other than being hereditarily indecomposable. After Knaster's thesis in 1922, there were only a few theoretical results concerning hereditarily indecomposable continua, and no really significant theorems or examples, until 1948. In that year, E. E. Moise, in a thesis written ] under the supervision of R. L. Moore, found a homogeneous (see p. 184) plane hereditarily indecomposable continuum, the pseudo-arc, with the property that it is homeomorphic to each of its non-degenerate subcontinua. This answered negatively the question posed by Mazurkiewicz in 1921 [88] as to whether every plane continuum homeomorphic to each of its non-degenerate subcontinua is an an are (that is, a homeomorph of I). However, Henderson showed in his thesis (1959) that in any metric space, any decomposable continuum that is homeomorphic to each of its non-degenerate subcon- tinua is an arc [42]. Hence, the conjecture of Mazurkiewicz was partially correct. 164 Bing, Moise, and others have extensively studied the pseudo-arc, and we shall present their findings later. For now, we give one method of constructing this continuum. A 22213 is a finite collection C of open (though not necessarily connected) sets (01,. .. ,cn) called lipkg such that cinej C C iff ji-j(£ 1. If each link has a diameter less than 6 , C is called an é-chain. If pecl, and qécn, then we have a chain from p to q. Chain D refines chain C if each link of D is a subset of a link of C. If D refines C in such a way that for each link 0 of C, the set of all links of D that lie in c is a subchain of D,then D is straight with respect to C. The basic terminology we need to describe the construc- tion of the pseudo-arc is that of "crooked chain." Consider a chain C = (01,. .. ,cn) from p to q. If11é4w then a chain D from p to q is 2221 crooked with respect to C if D is straight with respect to C. If n p5, then a chain D from p to q is very crooked with respect to C if D is a refine— ment of C, and D is the union of: (a) a chain from p to x5 Cn-l; (b) a chain from x to y5.02; (c) a chain from y to q, such that these chains are very crooked with respect to C-{cn}, C-[cl,cn}, and c-{bl[ respectively, and such that no two of them have in common any link that is not an end link of both of them. (If a chain C goes from p to q, then the 165 end links of C are the links containing p and q.) Note the similarity to Knaster's "method of bands" (Chapter 6). The above definitions are essentially as Moise gave them [97, pp. 581-583]. We can now define the pseudo-arc. Definition: Let 01’ C2, . . . be a sequence of chains from p to q such that: (a) C§ = U cl,j is a compact metric space; 3 (b) for each i, C is very crooked with respect to Ci i+l and Ci+1 is contained in the interior of Cf; (c) C contains five links; 1 (d) if c is a link of Ci and X is a subchain of Ci+1 which is maximal with respect to the property of being a subchain of Ci+1 and a refinement of the chain whose only link is c, then X consists of five links; (e) for each 1, each link of Ci has diameter less than l/i. Let M = n Cf; M is called a pseudo-arc [97, p. 583]. Theorem 12.1: If M, N are any sets satisfying the definition of a pseudo-arc, then they are homeomorphic. 166 3322;: See [97, p. 585]. Theorem 12.2: Every pseudo-arc is hereditarily indecompos- able [97, p. 585]. Outline of proof: M is clearly compact. Since we did not assume the chains have connected links, we must show that M is connected. Suppose it is the union of disjoint nonempty open sets H, K. Let i be such that 3/i< d(H,K). It follows that C1 is not a chain, which is a contradiction. Suppose N is any subcontinuum of M. We will use Janis- zewski's theorem (4.4) to show it is indecomposable. Let G; be the subchain of Ci consisting of all links of C1 that intersect N. Let K be any proper subcontinuum of N, and let I I 01' be the subchain of C1 consisting of all links of C1 which intersect K. It can be shown that for all but a I I I . _ finite number of integers, Ci - Ci contains two adjacent links of Ci. It follows from this, that for such i, the I . l I . set of all links of Ci+l which lie in links of Ci contains II two chains which "lie close together" such that one has Ci+l for a refinement. It follows from this that N-K C N. Thus, by Theorem 4.4, N is indecomposable. Moise comments [97, p.586] that his proof of M's being hereditarily indecomposable is quite similar to the corresponding proof in Knaster's thesis [59. p. 279]. We have also mentioned that Knaster and Moise used similar In fact, Moise suspected that Thus, methods of construction. their continua might be homeomorphic [97, p. 581]. 167 the following theorem of Bing should come as no surprise. It was published in 1951, three years after the appearance of the psuedo-arc. First, another definition. A metric continuum is chainable or snake-like (a term Bing credits to Choquet [8, p. 653]) if it can be covered by an E-chain for each 6 7 0. Theorem 12.3: If M, M' are non-degenerate compact metric continua that are hereditarily indecomposable and chain- able, then they are homeomorphic. Moreover, if p, q are in different composants of M,and p', q' are in different com- posants of M', then there is a homeomorphism of M to M. sending p to p' and q to q'. 3332;: See [10, pp. 44-45]. However, not all hereditarily indecomposable continua are homeomorphic to the pseudo-arc. In fact, Bing proved Theorem 12.4: There are as many non-homeomorphic plane hereditarily indecomposable continua as there are real numbers. 3332‘: See [10, p. 50]. Bing also extended Mazurkiewicz' results (Chapter 7) on the frequency ofoccurrence of indecomposable continua. In his monumental 1951 paper, Bing proved Theorem 12.5: Let S be En (n 02) or a Hilbert space. Then most continua are pseudo-arcs in the sense that if the set of all continua in S is given the Hausdorff metric, then the set of pseudo-arcs is of second category and in fact is a dense G6 set. 168 2322:: See [10, p. 46]. He extended this result in 1964. Recall that the links of a chain in E2 need not be connected, let alone open disks. However, Bing showed that the above theorem is true for pseudo-arcs constructed from chains with open disks for links: Theorem 12.6: Most (in the sense of Theorem 12.5) plane con- tinua are pseudo-arcs which for each 6 7C)can be covered with a linear chain whose links are open disks of diameter less than 6 . 131%: See [12, p. 122]. It might be conjectured by now that all indecomposable continua are at most one dimensional. In his thesis, direc- ted by G. T. Whyburn, J. L. Kelley proved (1940) that if there is a hereditarily indecomposable continuum of dimen- sion greater than one, then there is one of infinite dimen- sion [57, pp. 22-35]. However, the major result in this direction was proved by Bing in 1951: Theorem 12.7: There are infinite dimensional hereditarily indecomposable continua in a Hilbert cube and n-dimensional hereditarily indecomposable continua in En+l. More gen- erally, each (n+l)-dimensional continuum contains an n- dimensional hereditarily indecomposable continuum. 2322f: See [9, p. 270]. This is not merely an existence theorem: Bing's proof actually gives a way of constructing higher dimensional hereditarily indecomposable continua. We shall give that 169 construction in E3 after these definitions. An arc is E ~crooked if for each pair of points p, q there exist points r, s between p, q on the arc such that r lies be- tween p and s, and d(p,s)<’e , d(r,q)< 6: I , . 26 6 L9 9 7] Let a, b be two distinct points in E3. The desired continuum is to be the intersection of a decreasing sequence of bounded domains Di’ where Di 3 51+1; Di separates a and b; EB-Di has only two components, and no point of Di is more than l/i from either of them; and finally, each arc in D1 is 1/i crooked. D I! 1 {X5E3‘ min [1/4, d(a,b)/3]4d(a.x)(_ min [1/2, 2d(a,b)/3]}. Bing proves a general theorem which allows him to construct D2, D3, . . . satisfying the above conditions. Since Di 3 D then C = n D = n D 1+1, 1 hence i; C is a continuum by Theorem 2.1. C separates a from b, and 170 it separates E3 irreducibly into two complementary domains of C, by the third and fourth properties of the Di‘ Thus, by [45, pp. 98-99], the dimension of C is two. The last condition on the D1 gives the hereditarily indecomposability of C: Let K be any subcontinuum of C, and suppose K = AlJB, where A, B are proper subcontinua of K. Then there is an n70, p €A, qu such that d(p,B) 71/n, d(q,A) 71/n. Let U, V be connected cpen sets of Dn containing A, B respec- tively and such that d(p,V) 71/n, d(q.U)7 1/n. Let ernV, and consider an arc pxq in UlJV. Since d(p,xq)‘71/n, and d(px,q) 71/n, then nxq is not 1/n crooked. This is a contradiction [9, p. 268]. Although Bing and Moise proved that the pseudo-arc is homogeneous (we shall say much more about this at the end of the chapter), no such result is true for higher dimen- sional hereditarily indecomposable continua. In fact, Bing proved the following Theorem 12.8: If n is an integer greater than 1, then no n-dimensional hereditarily indecomposable continuum is homogeneous. pm: See [9, p. 272]. There are very few results in the literature charac- terizing hereditarily indecomposable continua in terms of other properties such as there are for ordinary indecompos- able continua. (c.f. Theorems 4.4, 4.11) However, we do have two theorems along this line. In 1929, Roberts and Dorroh answered a question of G. T. Whyburn [125] by proving 171 Theorem 12.9: A necessary and sufficient condition that a metric continuum C be hereditarily indecomposable is that no subcontinuum of C contain an irreducible separator of itself. pm: See [106, p. 61]. A much more recent (1966) result due to Zame is Theorem 12.10: A T2 continuum C is hereditarily indecom- posable iff for each pair of subcontinua, M, N, M-N is connected. Outline of proof: Suppose there exists a pair of sub- continua M, N such that M-N is disconnected: M-N = AlJB, where A, B are open in M—N, disjoint, and nonempty. Then by Lemma 4.2, NlJA, NlJB are continua. Hence NlJAlJB is a decomposable subcontinuum of C. Conversely, suppose M is a decomposable subcontinuum of C, say M = HlJK. If HIIK is connected, then it is a sub- continuum of C. (H u K)-(H n K) = [H-(Hn K)] u [K-(H nK)] is disconnected. The case of H11K disconnected is somewhat longer and will not be presented here. See [132, pp. 709-710]. Recall that a corresponding theorem for ordinary indecomposable continua (4.20) says that if C is a T2 indecomposable continuum and if K is any proper subcon- tinuum, then C-L is connected, where L is any subset of K. Just as for ordinary indecomposable continua, heredi- tarily indecomposable continua can sometimes be effectively represented by inverse limits. There always exists such a representation for metric continua of any sort, which is a l lllllJ 172 result first proved by Freudenthal [40]. See also [85, p. 149], and [47, pp. 75-76]. Conversely, Isbell proved in 1959 that if we are given an inverse sequence of compact subspaces of En, then the inverse limit is a subspace of E2n. Moreover, for n = 1, the hypothesis of "compact" may be dropped [47, p. 78]. McCord proved a related theorem in his thesis: The inverse limit of a sequence of compact metric spaces of dimension n 2n+1. We would like to start with such may be embedded in I an inverse sequence and know when the inverse limit is a hereditarily indecomposable continuum. In 1960, M. Brown established a criterion for this to happen. Let f: X——9Y, where X, Y are metric spaces, and let 67 0. Let L(E—,f) =sup{glx,yex, d(x,y)<6 =7 d(f(X).f(y))<6]. Suppose now that X is only assumed to be a topological space. f is 6-—crooked if for every path g: I~—>X, there c-t 4 1 such that |fg(0) - fg(t2)|4 e , exists t t Oétl- 2- 1' 2' and ‘fg(t1) - fg(l)] 4 6. Theorem 12.11: Let {Xi’fi j] be an inverse sequence of 7 locally connected metric continua with diameter di. Suppose é - for all n that fn,n+1 is n crooked, where én (1331?.-. L(2-ndi,fi’n_l)}. Then X“, is hereditarily indecomposable. 2323;: See [18, p. 130]. At this point, we seem to have exhausted the supply of structure theorems for general hereditarily indecomposable II|IIIIJ 173 continua. It would be interesting to know if such continua can be characterized in terms of irreducibility or some other property. Intuitively, a hereditarily indecomposable continuum ought to be "more irreducible" than an ordinary indecomposable continuum. However, there are some results of this nature for the pseudo-arc. In 1951, Bing gave the following characterization of the pseudo-arc (Theorem 12.14). Let p be a point of a metric continuum C such that for each 6 7'0, there exists an 6 —chain covering C such that only the first link of each chain contains p. Then p is an endpoint of C. (Under this definition, the only endpoint of Knaster's first semi-circle example is the origin [8, p. 662].) Lemma 12.12: Let C be as above; the following are equivalent: (a) Each non-degenerate subcontinuum of C containing p is irreducible between p and some other point of C. (b) If each of two subcontinua of C contain p, then one contains the other. 2222:: See [8, p. 661]. Lemma 12.15: Let C be a metric snake-like continuum. A point p613 is an endpoint of C iff it satisfies either a or b above. Proof: See [8, p. 661]. Theorem 12.14: A non-degenerate snake-like continuum is a pseudo-arc iff each point of it is an endpoint. Proof: See [8, p. 662]. We can restate the above results as follows. Let C be [I'lJ 174 a non-degenerate snake-like continuum. C is a pseudo-arc iff for each pé>C and for each non-degenerate subcontinuum K containing p, K is irreducible between p and some other point. Compare this with the case of an ordinary snake— 1ike indecomposable continuum, such as Knaster's first semi- circle example: Let p be any point except the origin and 3 let K = U Ci’ where 02 is the semi-circle containing p; 03 I is a semi-circle having an endpoint in common with C2. C1 is the other such semi-circle, provided C2 is not the semi- circle having (0,0) and (1,0) as endpoints; it is the empty set in this case. Then K is not irreducible between p and anything else. In contrast to the higher dimensional hereditarily indecomposable continua that Bing constructed, we have Theorem 12.15: The pseudo-arc does not separate the plane. Moreover, there exist plane hereditarily indecomposable continua which are not homeomorphic to the pseudo-arc that do not separate the plane. However, there exists a heredi- tarily indecomposable continuum which does separate the plane. Proof: Moise showed that the pseudo-arc is planar and homeomorphic to each of its non-degenerate subcontinua [97, p. 581]. However, in 1950, G. T. Whyburn had shown that no such continuum could separate the plane [126, pp. 519-520]. The second statement is due to R. D. Anderson [5, p. 185]. In fact, he announced that there exist in E2 uncount- ably many hereditarily indecomposable continua not sepa- [IIIIJ 175 rating E2 and not homeomorphic to the pseudo—arc, including one containing no pseudo-arc and another one, all of whose proper non—degenerate subcontinua are pseudo-arcs. Finally, Bing showed that there exists an example of the third type by constructing the example later known as the pseudo-circle [10, p. 48]. We shall say more about this example later. The pseudo-arc has other interesting properties with respect to the plane. Theorem 12.16: (a) There is a continuous collection of pseudo-arcs filling the plane. (b) There exists in the plane an uncountable set of disjoint continua, no one of which contains an are. 2392;: For (a), see R. D. Anderson [4, p. 550]. (b) This was first proved by R. L. Moore in 1928 [101, P. 86]. Of course, this was during the time when the only known example of a hereditarily indecomposable continuum was thatof"Knaster, which is homeomorphic to the pseudo—arc. So let 0 be a pseudo-arc. By the proof of Theorem 4.11, C has uncountably many disjoint composants. From each com- posant, select a subcontinuum of C contained in that com— Posant. This set of continua is uncountable, each two elements are disjoint, and none can be an arc. (C can be any hereditarily indecomposable continuum, of course.) We now consider some of the mapping properties of the Pseudo-arc. Shortly after Moise announced the discovery of the pseudo-arc, F. B. Jones asked 0. H. Hamilton whether m L—_____.__.___ 176 the pseudo-arc has the fixed point property with respect d- 0 continuous functions. That is, does the pseudo-arc C satisfy the condition that for every continuous function f taking C to itself there exists x530 such that f(x) = x? Hamilton showed that the answer is yes for not only the pseudo—arc, but for arbitrary snake-like continua. Theorem 12.17: Let Y Y . be a sequence of chains 1’ 2’ ' ‘ such that: (a) Y1 is a nonempty compact metric space, where Y1 is the closure of the set of points lying in the links of Y1; (b) for each i, Yi D Y;:I; (c) 133 giam(Yi) = 0, where diam(Yi) is the maximum diameter of the links of the chain Y1. Let M denote the continuum which is the intersection of the sets Yi’ If T is any continuous transformation of M into a subset of itself, then there exists a p6 M such that T(p)==p. 2322;: See [41]. At the Summer Institute on Set-Theoretic Topology, 1955, R. H. Bing raised the question of what characterizes the continuous images of the pseudo—arc. In other words, is there an analog for the pseudo-arc of the Hahn-Mazurkiewicz theorem for the arc. (This theorem says that a metric con— tinuum C is a continuous image of an arc iff C is locally connected [44, p. 129].) One result in this direction is Theorem 12.18: Every snake-like continuum is a continuous image of a pseudo-arc. 177 2322:: J. Mioduszewski proved this in 1962 [96] using inverse limits. He also remarked that it seemed to follow from one of Bing's theorems [8, Theorem 5] and one of Lehner's [79, Theorem 1]. (G. Lehner was a thesis student of Bing.) L. Fearnley also proved this theorem in 1964 [33, p. 389]. The first characterization seems to have been given by A. Lelek in 1962, using the following terms. A wgak ghain in a metric space is a finite sequence of sets Xl" .. , X m such that Xiflxj £ ¢ if li-jl 51. Note that the X1 are not assumed to be either open or connected. Moreover, Xiflxj # ¢ does not imply li-jl $14 A weak chain {Xi}m‘ is a refine- I 'I ment of a weak chain {Y3} provided that each Xi is con— l tained in some Y. such that 'j. - jklél.if li-kiéld Ji 1 Finally, a continuum C is weakly chainable provided there exists an infinite sequence {Ci} = {(01,5hff} of finite open covers of C such that each G1 is a weak chain, each link of G1 has diameter less than 1/1, and Gn+1 is a refine- ment of Gn' Theorem 12.19: A metric continuum is a continuous image of a pseudo-arc iff it is weakly chainable. 3392;: See [80, p. 274]. Note that Theorem 12.18 follows directly from 12.19, since if C is chainable, it is weakly chainable. L. Fearnley also established some characterizations, using the following terminology. A p-chain is a finite 178 sequence of sets such that each, except the last, intersects its successor. (c.f. "weak chain") If P = (p1,. .. ,Pn) and Q = (q1" .. ,qm) are p-chains and each link pi of P is a subset of a link qx of Q, then the sequence of ordered i pairs {(i,xi)} is a pattern of P in Q. If ‘Xi - lefil whenever [i-j|.51, 15 i,jg¢n then the pattern is an 3: pattern of P in Q. If a p-chain P = (p1,. .. ,pn) has an r—pattern of the form (l,xl = 1), (2.x2),. .. ,(n,xn = m) in a p-chain Q = (ql" .. ,qm) then P is a normal refinement of Q. Finally, let H be a closed connected separable metric space. H is p-chainable if there is a sequence of p-chains P1,. .. such that for each i: (a) the union of the elements of Pi is H; (b) Pi+l is a normal refinement of Pi; (c) the diameter of each link of Pi is less than l/i; (d) the closure of each link of Pi+1 is a subset of the link of Pi to which it corresponds under the r-pattern of Pi+l in Pi. Theorem 12.20: (a) In order that H (as defined above) be a continuous image of the pseudo-arc, it is necessary and suf- ficient that H be p-chainable. (b) A metric continuum C is a continuous image of the pseudo-arc iff C is p—chainable with p-chains whose links are open sets. Proof: For (a), see [55. p. 587], and for (b). see [33, p. 588]. 179 Theorem 12.21: (a) Let K be a chainable separable closed connected metric space. Then K is a continuous image of the pseudo-arc. (b) The class of continuous images of the pseudo-arc and the class of continuous images of all such sets as K are identical. 2322:: For (a), see [55, p. 589]. Also compare (a) with Theorem 12.17. (b) is a corollary of (a). These results still do not fully answer the question of whether there is an analog of the Hahn-Mazurkiewicz theorem for the pseudo-arc. However, Fearnley went on to show that there is no characterization of the continuous images of the pseudo-arc in terms of local properties by constructing locally homeomorphic metric continua H and K such that H is a continuous image of the pseudo-arc and K is not [33, pp. 591-395]. It might be conjectured that using inverse limits to describe the pseudo-arc would be easier than using chain conditions. However, in most cases there are infinitely many different binding maps, so that the situation is not greatly improved. In 1964, Henderson was able to construct the pseudo-arc as an inverse limit of a sequence of arcs and one binding map. Roughly speaking, the map was obtained by taking f(x) = x2 on I and "notching its graph with an infinite set of non-intersecting,yr's which accumulate at (1,1)." [45, p. 421] See the figure on the next page. The proof of this function's existence may be found in the 180 (1.1) (0.0) paper cited above. (See also the Math Reviews 29-4059 for some comments about errors.) The pseudo-arc is preserved by inverse limits and mono- tone maps. Explicitly, Bing has shown Theorem 12.22: Let M denote the pseudo-arc and let N be any non-degenerate monotone continuous image of M. Then M and N are homeomorphic. 2332;: See[10, p. 47]. Theorem 12.25: Let {Xi’ fi,j] be an inverse sequence of pseudo-arcs, and let X“, be the inverse limit. If X“ is non-degenerate, then it is a pseudo-arc [105, p. 599]. 2229:: By Theorem 11.2, Xw is a hereditarily indecom— posable continuum. Reed has shown that X“, is snake-like [105, p. 598]. Therefore, by Theorem 12.3, Xag is a pseudo-arc. We now discuss one last example of a hereditarily indecomposable continuum. In 1951, R. H. Bing described a plane non-snake-like circularly chainable hereditarily 181 indecomposable continuum. which has since become known as a pseudo-circle. (A.metric continuum is circularly ppginable if it can be covered for each 6 7C>by an 5'-chain whose first and last links intersect each other.) Bing described his example this way [10, p. 48]. Let 2 D such 1’ D2,. .. be a sequence of circular chains in E that: (a) each link of D1 is an open circular disk of diame- ter less than l/i; (b) the closure of each link of D1+1 is contained in a link of Di; (0) the union, Ai’ of the links of D1 is homeomorphic to the interior of an annulus; (d) each complementary domain of A1+1 contains a com- plementary domain of Ai; (e v if E1 is a proper subchain of Bi and E1+1 is a proper subchain of Di+l contained in Ei’ then E1+1 is very crooked in E1. M = ?)Ai is called a pseudo-circle. Bing proved that such sets exist and that they separate the plane. Fearnley pointed out that every proper non-degenerate subcontinuum of it is a pseudo-arc [54, p. 491]. After defining it, Bing asked if all such continua are homogeneous and whether they are homeomorphic. Fearnley has recently (1969) answered these questions, as well as some others. Theorem 12.24: (a) The pseudo-circle is unique in the sense 182 that any two continua satisfying the above definition are homeomorphic. (b) The pseudo-circle is not homogeneous. 3322:: See [55, pp. 598-401] or [57] for a proof of (a). Fearnley's proof of (b) may be found in [56]. J. T. Rogers Jr. also proved (b) [107] in a thesis supervised by F. B. Jones. Fearnley also investigated some mapping properties of the pseudo-circle. We need the following definition in order to state his result. A circular p-chain is a p-chain in which the first and last links intersect. Theorem 12.25: In order that a continuum C be a continuous image of a pseudo-circle, it is necessary and sufficient that C be circularly p-chainable. 3323;: See [34, p. 507]. Theorem 12.26: (a) Every plane circularly chainable con- tinuum is a continuous image of the pseudo-circle. (b) Every snake-like continuum is a continuous image of the pseudo-circle. 3322;: See [34, p. 510] for (a) and [34, p. 512] for (b). Thus, the pseudo-arc is a continuous image of the pseudo-circle. It is not known if the converse is true. However, Fearnley has indicated that he has a paper in progress which answers this question as well as whether every solenoid is a continuous image of the pseudo-circle. We have mentioned that the pseudo-arc is homogeneous, 185 while the pseudo—circle fails to have this property. The homogeneity of the pseudo-arc has an interesting history which is closely related to that of finding all homogeneous plane continua. Consequently, we shall discuss this prob- lem in some detail. We begin with some developments in the early Polish school of mathematics. Sierpinski formulated the definition of homogeneity in a paper which appeared in the first volume of the Fundamenta Mathematicae [111, pp. 15-16]. In the same issue, Knaster and Kuratowski posed the question of whether every non- degenerate homogeneous plane continuum is a simple closed curve (that is, a homeomorph of SI) [61]. Mazurkiewicz proved in 1924 that the answer is yes if the continuum is also assumed to be locally connected [89]. During the years between 1924 and 1948, two false solutions were published. Of course, it was not known that these solutions were false until Bing and Moise showed that the pseudo-arc is homogeneous. Waraszkiewicz announced in 1957 that all non-degenerate homogeneous plane continua are simple closed curves [124]. Choquet's paper [24] went even further in 1944. In it, he asserted that every compact homogeneous plane set is either: (a) a finite set of points; (b) a totally disconnected perfect set; (c) homeomorphic to the union of a collection of con- centric circles such that the intersection of this union and a line through the center of the circles 184 is either a finite set or a totally disconnected perfect set. In 1949, F. B. Jones proved that under slightly stronger hypotheses, Waraszkiewicz' theorem is correct. We need the following definition in order to give a precise statement of Jones' result. A continuum C is aposypdetic at x if for each point y6(L-fiq, there exists a subcontinuum K of C and an open set U of C such that C-{y} D K D U D {x}. Jones gives the following explaination of his term "apo- syndetic": In Greek, "apo" means "away from", "syn" means "together", while "deo" signifies "to bind". Thus, the word "aposyndetic" means "bound together away from" [51, p. 546]. Theorem 12.27: Let C be a non-degenerate homogeneous plane continuum. If C is either aposyndetic at all points or if no point of C is a cut point, then C is a simple closed curve. 3323:: See [52]. Jones also suggested that Waraszkiewicz' error may have been to confuse the idea of a cut point of a continuum with that of a separating point [54, p. 66]. Shortly after seeing Moise's pseudo-arc, R. H. Bing proved that it is homogeneous. Moise gave his own proof shortly thereafter. Theorem 12.28: The pseudo-arc is homogeneous. Epppf: See [7] for Bing's(l948)proof, and [98] for Moise's proof (1949). In view of the results of Warszkiewicz and Choquet, it 185 is not surprising that some people questioned the homogene- ity of the pseudo-arc. Isaac Kapuano presented a paper in 1955 in which he claimed that the pseudo-arc is not homo- geneous [55]. He noted that it would be interesting to know exactly what part of Bing's paper is contradicted by his work. However, an error was discovered in his own work, so later in 1955. he published an attempt to correct it [56]. Moreover, neither paper received much criticism in the Mathematical Reviews [Math Reviews 15: 146, 555]. However, mathematicians seem more inclined to accept the results of Bing and Moise than those of Kapuano. Thus, what might have developed into a "lengthy debate" just faded away. Theorem 12.29: Each non-degenerate homogeneous snake-like continuum is a pseudo-arc. Thus, a non-degenerate snake- 1ike continuum is a pseudo-arc iff it is either hereditarily indecomposable or homogeneous. 2322:: For the first statement, see [11]. The second statement is a summary of Theorems 12.2, 12.5, 12.28, and the first statement of 12.29. Knaster and Kuratowski's question can now be expanded to the problem of finding all homogeneous plane continua. F. B. Jones gave the following classification of possible homogeneous plane continua [54, p. 67]: (a) those which do not separate E2; (b) those which are decomposable; (0) those which separate E2 and are indecomposable. This is a reasonable approach, in view of one of Jones' 186 earlier theorems: Theorem 12.50: If C is a homogeneous plane continuum which does not separate the plane, then C is indecomposable. M: See [53, p. 859]. At the time Jones gave his classification, a point and the pseudo-arc were the only known non-homeomorphic examples of type (a). A simple closed curve and an example dis- covered simultaneously by Bing and Jones, called a circle of pseudo-arcs [15] were the only known examples