I“I II‘ I I, III II II: II III III II I I I I I II I I II II II —I _. I I N 'moucn I I w I II I I 00 \I (INSTRUCTION 0F ANTOINE-TYPE MAPS BETWEEN CONTINUA IN EUCLIDEAN SPACE Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY JAMES MICHAEL SOBOTA 1970 IJBRARY Michigan State University *- THt-‘cgvs -w— This is to certify that the thesis entitled CONSTRUCTION OF ANTOINE-TYPE MAPS BETWEEN CONTINUA IN EUCLIDEAN SPACE presented by James Michael Sobota has been accepted towards fulfillment of the requirements for E. h.D. degree inJaLhematics Date AHEUS’C 7. 1970 0-169 I I l: - BINDING BY IIIIAIS & SIIIS' 890K BINIIEIIY III . a nuumr nIIDERS ABSTRACT CONSTRUCTION OF ANTOINE - TYPE MAPS BETWEEN CONTINUA IN EUCLIDEAN SPACE By James Michael Sobota In this paper we consider modifications of the general extension problem prOposed by Antoine. We restrict our attention to continuous functions defined on En, taking one given continuum to another. In general we work with the following definition: Definition 2.1 Two continua A and B in En are weakly . . . . - n n equivalent if there ex1sts a continuous t: E a E such that 1.) f(A) = B 2,) f(En - A) = E“ - B. Theorem 2.4 Any non-separating continuum ACZE3 is weakly equivalent to a point. Theorem 2.7 Let W be a wild arc in E3 such that W = ALJB where A and B are tame arcs in E3 and AFIB = p. Then W is weakly equivalent to a tame arc. 3 Theorem 2.9 Let W be a wild arc in E such that W lies on the boundary of a 3-cell in E3. Then any tame arc A in 3 E is weakly equivalent to W. Theorem 2.11 Any arc W which is the union of a finite number of tame arcs is weakly equivalent to any arc which lies on the James Michael Sobota 3 boundary of a 3-ball in E or which lies on the boundary of a 2vdisc in E3. In Chapter IV we generalize some of these results to higher dimensions. We have the following theorem: k k+3 Theorem 4.2 For any k>>0, there exists a wild k-cell W c:S + which is weakly equivalent to a tame k-cell Tk<: Sk 3. In fact Wk goes homeomorphically onto Tk. CONSTRUCTION OF ANTOINE - TYPE MAPS BETWEEN CONTINUA IN EUCIIDEAN SPACE BY James Michael Sobota A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1970 C7- c.5709 / ' J9'7“ "7/ ACKNOWLEDGEMENTS The author wishes to express his gratitude to Professor P.H. Doyle for suggesting the problems discussed in this thesis and for his interest and direction during the past two years. He thanks the faculty and graduate students of the Department of Mathematics, with whom he has had many interesting discussions during the course of this study; and his wife, Marcia Anne, for her help and encouragement. ii CHAPTER II III IV VI INTRODUCTION TABLE OF CONTENTS OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO WEAK EQUIVALENCE BETWEEN CONTINUA ................ PROPER EQUIVALENCE OF SETS . . . .................... WEAK EQUIVALENCE BETWEEN k-CELIS IN S .......... A MAP ON E3 PRESERVING COMPLEMENTS a.e . ....... SOME POINT SET RESULTS AND PATHOIDGY ............. BIB LIOGRAPHY OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO iii 17 19 21 23 27 LIST OF FIGURES iv 11 15 16 22 CHAPTER I INTRODUCTION The problems we are concerned with in this paper arise from the problem Antoine discussed in [2]. He asked whether given a homeomorphism between compact sets in E“, does the homeomorphism extend to E“, or if not, does it extend to a homeomorphism on a neighborhood of the domain set. Antoine answered his question negatively via an example in E3. The Antoine Necklace is a cantor set in E3 lying on an arc and the arc has the property that a homeomorphism of this arc onto a segment in E3 does not extend to a homeomorphism of E to itself. Fox and Artin in [9] give more examples, among them examples of arcs in 33 such that homeomorphisms of these arcs onto segments in E3 extend only to tapered neighborhoods of the arcs (see also Persinger [14]). The extension problem for 2-spheres in E3 was answered negatively by Alexander in [1], the Horned Sphere of Alexander being the example. We form several generalizations and restrictions of Antoine's problem, we give solutions and partial solutions in the later chapters. We first ask the general question: given pairs (X,A) and (Y,B), when can a map f: A a B (onto) be extended to an epi-map f:.X a Y such that f takes X - A onto Y - B. We give examples showing simple cases where this need not happen. Example 1.1 Consider the pairs (X,A) and (Y,B) where X = {0,1,2,3}, A 8 {0,1}, Y = {O,1,2,3,4}, B = {0,1} and all Spaces are discrete. Let f: A a B be defined by f(x) = x. f cannot extend to an epi-map F:.X ~‘Y Such that X - A goes onto Y - B. But this example is somewhat trivial due to the fact that the cardinality of Y is greater than that of X. Example 1.2 Let X = Y = 2+ (the positive integers), A ' {3,4,5,...}, B = {6,7,8,...}. Let f: A a B be defined by f(x) - x + 3. Again f cannot extend since Y - B has greater cardinality than X - A. We next give an example where the trouble is not caused by cardinality differences. The following theorem is from Hempel [10]. Theorem 1.3 Let M be a closed, connected 2-manifold which is tamely embedded in E3 and let f be a map of E3 onto itself such that f restricted to M is a homeomorphism and f(E3 - M) ' E3 - f(M). Then f(M) is tamely embedded in E3. Example 1.4 Let X = Y = E3 and let A = 82, B - A2 the Alexander horned sphere. Let f: A a B be a homeomorphism. By Theorem 1.3, we know that if f extends to F: X a Y such that F(X - A) = Y - B then B is tame. But we know A2 is wild in BB, hence no extension exists. Example 1.5 Let X = [0,»), A = [a,b], Y = L (L is the "long line", see Hocking and Young [11] p. 55), B = [a,b], where [a,b] is a nice interval in L. Let f: A a B be defined by f(x) = x. f cannot be extended to a map from X onto Y be- cause Y is "too long". We do know however that certain epi-maps extend to epi- maps preserving complements. If A and B are equivalently embedded in E3 we can extend homeomorphisms. The following theorem and example also tell us more. Theorem 1.6 Let A # ¢ be a proper open subset of X. There exists a pair (Y,B) such that any map f: A.4 B extends to an epi-map F: X 4'Y such that X - A is mapped onto 'Y - B. Proof: Let Y be Sierpinski Space, B the open point. If A is proper and closed we have the same result. Example 1.7 The pair (E1, [0,1]) has the property that if [0,1] is mapped onto a non separating continuum B GIMP, an n-manifold, in such a way that the images of {O} and {1] are arcwise accessible from MP - B, then the map extends to a map from E1 onto MP which preserves complements. The illustration of this fact is found in the proof of Theorem 2.9. We might also ask whether a homeomorphism between arcs n . . . on spheres in E extends to a continuous function preserv1ng complements. This is given a negative answer in Example 1.8. A similar question for 2-spheres is found to have a negative answer in Example 1.4. Example 1.8 Let T be a tame are on 82, a tame 2-Sphere in E3 and let W be a wild arc on the horned sphere of Alexander, 2 A . Let H: 32 a A2 be a homeomorphism extending the homeo- morphism h: T «IW. If we assume H extends to H: E3 a E3 2 2 such that [-1033 - S ) - E3 - A , then by Theorem 1.3, A2 is tame. We know this is false. Another possible investigation is that of connected functions. A Connected function is a function (not necessarily continuous) which preserves connected sets. We could ask whether an epi-map f: A d B extends to a connected function. This is a weakening of the problem since a continuous function is connected but not conversely. Example 1.9 A connected function which is not continuous. Let X - [(x,y) : 0 s x s 1, -l s y s l} and let Y = {(x,y) : 01< x-s 1, y = sin £1 U {(0,y) : -l s y S 1}. Define f: X a Y by f(x,y) = (x, sin i), x > 0 {(0,0), x = 0 f is obviously not continuous but it is connected. In the remaining chapters we discuss more aSpects of the extension problem. In Chapter III we discuss the extensions of prOper maps. We give some "folklore results” and use these to obtain some answers to extension problems. The main emphasis of Chapters II, IV, and V is the actual construction of maps. In Chapter II we ask whether there exists a map f: E3 d E3 such that f(A) = B and f(E3 - A) - 33 - B. We construct some maps f for some Specified continua A and B. In the last part of Chapter II we weaken our requirements and construct maps between complements of non-separating continua in E3. In Chapter IV we generalize Chapter II to higher dimensions, and construct a map from Ska."3 onto Skr'”3 taking a wild k-cell onto a tame one. Chapter V contains a different approach to the construc- tion problems in Chapters II and IV. We construct functions which map wild arcs onto tame arcs but preserve complements only modulo a set of 3-dimensional Lebesgue measure zero. In Chapter VI we consider some new definitions related to those of Chapter III and show some results dealing with more general point-set topology. We also consider some pathology in measure and dimension. In this paper we will adopt the following definitions and notation. By En we will mean n-dimensional euclidean Space with the usual topology, and Sn will denote the unit n-Sphere. )1X2+x2+...x2 =1}c:E n+1 1 2 n+1 ' n S {(xl, x2,...xn+1 An embedding is a homeomorphism of a Space A into a Space B, and an arc is the homeomorphic image of the closed unit interval. A and B are equivalently embedded in Sn if there exists a homeomorphism of Sn onto itself taking A onto B. We adopt the definitions of wild_and £322.33 found in Fox and Artin [9]. The piecewise linear topology discussed may be found in Hudson [12], and the dimension theory in Hurewicz and Wallman [13]. By m1(A) we mean the i-dimension LebeSgue measure of the set A, as can be found in Rudin [16]. Our goal in this paper is not necessarily the most gen- eral result possible, but rather constructive results which depend on the positioning of the continua A and B in En. CHAPTER II WEAK.EQUIVAIENCE BETWEEN CONTINUA In this chapter we answer some-modifications of Antoine's 3 question in E . We concern ourselves primarily with the con- struction of continuous functions. We make the following defini- tion. Definition 2.1 Two continua A and B in En are weakly equivalent if there exists a continuous f: En a En such that IJ fm)=B 2.) £03“ - A) = En - B. Examples Any two points are weakly equivalent. In fact, any two equivalently embedded continua are weakly equivalent. The 3 solid flat torus in E is weakly equivalent to an unknotted circle, however they are not equivalent. In the remainder of this chapter we restrict our study 3 . to non-separating continua in E . Theorem 2.2 18 a corollary to Theorem 2.4 but a simpler technique suffices so we include the proof. 3 Theorem 2.2 Any tame arc A in E is weakly equivalent to 3 a point b' in E . . . . 3 Proof: ‘We construct the dealred continuous function f: E a E as follows. We first assume that A is a straight segment, for if not there exists an E3 homeomorphism taking A onto a straight segment. Let b be the midpoint of A. We enclose A in a decreasing sequence of closed cubes {Ci} converging down to A. Similarly we enclose the point b' in a decreasing sequence of closed cubes {Ci} centered at and converging to b'. Con- sider the collection {E3 - Ci} of open sets which form an open covering of E3 - A. We construct a family {f1} of continuous functions, each fi will be defined on Cl(E3 - Ci) and hence also de- fined on E3 - Ci' There exists f1 : C1(E3 - C1) a Cl(E3 - Cl)' We extend a 3 3 l 3 f1 to f2 . Cl(E - C2) a C1(E C2), so that f1 f2 on the intersection of their domains. We construct f2 as follows. For each x e Fr(C1) we map the part of the straight line segment 3;. lying in Cl(C1 - C onto the part of the segment 2) b'f1(x) lying in 01(01 - C5). See Figure l. «x C \B -c Figure 1. We continue in this way, extending each . 3 3 , . 3 3 , fi' C1(E - Ci) fl C1(E - Ci to fi+l° C1(E - Ci+l) a C1(E -Ci+1). We have thus constructed a collection of maps {fi} with domains {E3 - Ci} such that the maps agree on the inter- sections of their domains. There exists a unique continuous f: E3 - A a E3 - b' which extends the fi's. We extend f to a map defined on 33 (we denote the extension by f also) by defining f(A) = b'. This gives the required function. lfimmngLgi Let A be a non-separating continuum in E3- Then there exists a decreasing sequence of compact 3-manifolds with connected boundary {Mi} such that nMi = A, M <: Int Mi' 1+1 3 Proof: We first triangulate E . Let M, be a second derived 1 neighborhood of the set of simplicies meeting A in E3. For each i let M1+1 be such a second derived neighborhood in Mi' We claim that the {M1} just defined is the required collection of manifolds. By Theorem 2.11 in Hudson [12], the M1 are all regular neighborhoods, hence we have that each Mi is a compact '3-manifold, and since A does not separate, we assume each M has a connected boundary. 1 If we require the mesh of the subdivisions to go to zero, we have A - “My Theorem 2.4 Any non-separating continuum ACE3 is weakly equivalent to a point. 10 Proof: By Lemma 2.3 we enclose A in a sequence of compact 3-manifolds with connected boundary {Mi} such that Mk+1<: Int Mk. 'We enclose the pOint b in a sequence of closed 3-balls {Bk} centered at and converging to b. Let N - C1(Mk - Mk+l) which is a compact 3-manifold k with two boundary components, aMk and 3Mk+1. Let Ak = C1(Bk - Bk+l)’ a closed anular region with boundary com- ponents 5BR and 5Bk+1. We map Nk onto Ak as follows. Remove Open 2-discs Uk and Uk+l from aMk and 3Mk+1 respectively, and remove open 2-discs VR and Vk+1 from 53k and BB Construct tubes from aUk to 5Uk+l k+l' k+l° Let Ck be the closed 3-cell and part of the constructed tube. Let and from aVk to 3V bounded by Uk’ Uk+1 be the closed 3-cell bounded by V and part of ' Ck the constructed tube. k’ Vk+1 There exists a homeomorphism gk from Ck onto Ci and hence a homeomorphism of aUk onto aVk. Similarly we have a homeomorphism gk+1 : aUk+1 ~ 3Vk+1. Since gk is defined on a closed subset of 'aMk - Uk into aBk - Vk’ a 2-cell, we can extend gk by Tietze's Extension Theorem to: Gk : (3M1 - Uk) a (33k - Vk), Similarly g1“.1 extends to Gk+l' These maps give rise to a map fk from (3Mk - Uk) U (3Mk+1 - Uk+l) U {Z-dimensional boundary of into a 3-cell. Again by Tietze's Extension Ck ' (UkU Uk+l)} Theorem, fk extends to Fk : Nk.fi Ak' 11 If the Uk and Vk are chosen properly for each k, 3 3 we have a family of open sets {E - Mk] covering E - A and a family {FR} of maps such that Fi = FJ on Ni n N Thus j' there is a unique extension F : E3 - A a E3 - b. We again extend to a map on E3 by setting F(A) = b. The result is the required map. The following corollary is a special case of Theorem 2.4. We are restricting our study to arcs in the next several results. 3 Corollary 2.5 Any arc A in E is weakly equivalent to a point. We now extend to show certain arcs are weakly equivalent to others. We begin by giving an example of a wild arc which is weakly equivalent to a tame arc in E . Example 2.6 Consider the wild arc W in Figure 2. A and B are tame arcs intersecting at p. We can map W onto B by shrinking A to p. This can be done by a map defined on E3 which preserves complements. B QG—G-em. A Figure 2 12 This example generalizes to Theorem 2.7. Theorem 2.7 let W be a wild arc in E3 such that W = A U B where A and B are tame arcs in E3 and A.n B = p. Then W is weakly equivalent to a tame arc. Proof: Since A is tame in E3, there exists a homeomorphism 3 3 . 3 h: E a E such that h(A) is a Straight segment in E . Since B is tame, h(B) is also tame and h(A) n h(B) = h(p). By Example 3.6 we can shrink h(A) to h(p). Corollary 2.8 Let W be a wild arc in E3 where ‘W = A1 U A2 U°°°U Ah where A1 is a tame arc in E3 and A1 n A1+1 = pi. Then. W is weakly equivalent to a tame arc. Proof: Apply the proof of Theorem 2.7 n times. We now give some constructive theorems to show that tame arcs are weakly equivalent to certain wild arcs. Hence we get a weak equivalence between certain types of wild arcs. Theorem 2.9 Let W be a wild arc in E3 such that W lies on the boundary of a 3-ce11 in E3. Then any tame arc A in E3 is weakly equivalent to W. Proof: We define the map f: E3 a E3, such that f(A) =‘W and f(E3 — A) I E3 - W, by the composition of three maps. We first assume A lies on the x-axis in E3. We define f1 : E3 4 E: = f(x,y.2) = z 2 0} by l3 (X.y.2) z 2 0 f1(x,y,2) = { (x,y,-z) z < 0 . We note that f1(A) = A. Let -X = B3 U L where B3 is a closed 3-ball and L 3 .E+ 2. noting that f2 can be chosen so that f2(A) EBB3 - {0}, 3 is a ray with end point {0} in BB . We define f a X preserving complements. 3 3 Assume W CZC where C is a closed 3-cell in E3. We construct f3: X a E3 using a collection of maps. There is a homeomorphism ho : 33 U [0,1] a C3 U K where K is an arc meeting C3 at only one end point and ho (A) ==W. We now map the "tail" of L onto the remainder of E3 by re- peated applications of the Hahn-Mazurkiewicz theorem, Hocking 3 . and Young [11] p. 129. We write E - C3 =UFi where each Fi is compact, connected and locally connected and Fi<: Int Fi+l° We extend ho to h by defining: l 3 1 = ho on B U [0,1], h1 maps [3/2, 2] onto F1 and h1 maps [1, 3/2] onto the path from ho(l) to h1(3/2). We extend h1 to h2 by defining: h 3 hz - hl on B u [0,2], h2 maps [5/2, 3] onto F2 and h2 maps [2, 5/2] onto the path from h1(2) to h2(5/2). to h by defining: We continue extending each h k k-l h = h 3 k k-l on B U [0,k], l4 2k+1 hk maps [-§-, k+l] onto Fk and 2k+1 2k+l hk maps [k, 2 ] onto the path from hk_1(k) to hk( 2 ). This gives a map f3: X a E3. Finally we define f = f f f . 3 2 1 Theorem 2.10 Let W be a wild arc in E3 such that W lies on the boundaryrof a 2-disc. Then any tame arc A.C:E3Iois weakly equi- “valent to W. Proof: The same technique used in the proof of Theorem 2.9 suffices here. We now combine Corollary 2.8 with Theorems 2.9 and 2.10 to show certain wild arcs are weakly equivalent. Theorem 2.11 Any arc W which is the union of a finite number of tame arcs is weakly equivalent to any are which lies on the boundary of a 3-ba11 in E3 or which lies on the boundary of a 2-disc in E3. We define two continua A and B in En to be 2331 weakly equivalent if there exists a map f from En - A onto En - B. In [6], M. Brown defined the concept of cellularity in a manifold. A set A in an n-manifold is cellular if there is a decreasing sequence of n-cells {Ci} such that C Cilnt Ci and A ={1Ci. A set is pointlike i+1 it it has the same complement as a point. It is known that a set is pointlike if and only if it is cellular. 15 Remark Any two cellular sets in En are very weakly equi- valent. The converse is false as seen in Example 2.13. We do however have the following. Theorem 2.12 Any non-separating continua A in E3 is very weakly equivalent to any cellular set B in E3. Proof: We mimic the proof of Theorem 2.4. Enclose A in a decreasing sequence of 3-manifolds and enclose B in a de- creasing sequence of 3-cells. We construct the map exactly as in the proof of Theorem 2.4 except we omit the last extension. Example 2.13 There is a non-cellular arc A E3 which is very weakly equivalent to a cellular arc B. We use Example 1.1 in Fox and Artin [9]. The complement of this arc is not simply connected hence the arc is not cellular. But by Theorem 2.12 the arc is very weakly equivalent to a segment. Figure 3 We may extend Theorem 2.12 to non-separating continua in En using techniques similar to those used in the proof of Theorem 2.9. We first prove the more general result. 16 Theorem 2.14 Any two open connected n-manifolds are continuous images Of each other. Proof: Let M1 and M2 be Open connected n-manifolds and let L be a locally flat ray in M1 such that L is closed in M1. Enclose L in a product neighborhood En-1 X [0,1) as in Figure 4. Figure 4 '_ [1‘]. o 11"]. Let M1 - C1(E X [0,l)). There is a copy of E in Mi and Mi retracts onto this copy Of En-l. There is f-M'-~Ii:n-1 f etdta i-a 1. 1 . 1 x en 3 O n ep m p f2: M1 4 En.1 x [0,1). We also have a retraction f : E“'1 x [0,1) 4 L. Construct a map f4 : L 4 M.2 using the Hahn-Mazurkiewicz Theorem as in the proof Of Theorem 2.9. Define f = f4 f3 f2 : M1 a M2. . . . n Theorem 2.15 Any two non-separating continua in E are very weakly equivalent. . . 1'1 Proof: The complements of non-separating continua in E are Open connected n-manifolds. CHAPTER III PROPER EQUIVALENCE OF SETS Let f: X a Y be a continuous surjection. f is called 1 a proper map if for each compact A<:‘Y, f- (A) is compact in X. We can modify the general Antoine type problem in terms of proper maps. Given sets in En when does there exist proper maps between them and when do these maps extend to proper maps on En? We first state some known results on proper maps. Lemma 3.1 Let M1 and M2 be connected n-manifolds and let f: M1 d M2 be a proper map. Then f extends to a map from the one point compactification of M onto the one point 1 compactification of M2. Lemma 3.2 If M1 and M2 are connected n-manifolds and f: M d M is a proper map, then M has at least as many 1 2 ends as M2, Ie(M1)\ 2 \e(M2)I. 1 Lemma 3.3 If F: M1 ~ M2 is a proper map, f extends to a map from the Freudenthal compactification Of M1 to that of M2. Definition 3.4 Let M1 and M2 be connected n-manifolds. M1 and M2 are prpperly equivalent if there exists proper maps f1 : M1 a M2 and f2 : M2 a M1- l7 18 Theorem 3.5 Let A be a convergent sequence Of points in SD n and let B be a cantor set in Sn. S - A and Sn - B can- not be properly equivalent. Proof: If f: Sn - A « Sn - B were proper, then by Lemma 3.2 |e(Sn - A)| 2 \e(Sn - B)‘. But Sn - B has an uncountable number of ends and Sn - A has only countably many. The above method of proof shows that if A and B are compact and B has more components than A then Sn - A is not properly equivalent to Sn - B. Theorem 3.6 If M1 and M2 are pr0perly equivalent, then they have the same number of ends. The following theorem was proved by Fort in [8]. We give a much simpler proof here. Theorem 3.7 Let Cl and C2 be cantor sets in E“. If En - C1 is homeomorphic to En - C2, then C1 and c2 are equivalently embedded. n Proof: Let h: E - C1 4 En - C2 be a homeomorphism. Then h is proper and by Lemma 3.3 h extends to H : En 4 En which is a homeomorphism. Theorem 3.8 Let A and B be cellular continua in S“. Then n S - A and Sn - B are prOperly equivalent. CHAPTER IV WEAK EQUIVALENCE BETWEEN k-CELIS IN sn Chapter IV extends some results of Chapter II to higher dimensions. Using Theorem 2.4 with E3 replaced by 83, we construct a map defined on S4 which takes a wild arc to a tame arc and preserves complements. We use an easy induction to Show a wild k-cell in sk+3 is weakly equivalent to a tame k-cell. We lower the dimension to k+2 using Corollary 2.8 with E3 replaced by 53. Theorem 4.1 There exists a wild arc W CZS4 which is weakly 4 equivalent to a tame arc T C18 . 3 Proof: A wild arc A in S is weakly equivalent to a point in S3 by Corollary 2.5. Thus there exists f: 83 a 83 such 3 that f(A) = b, f(S3 - A) = S - b, and n1(33 - A) # 1. We form the quotient space S3/A by the natural map 3 l p : S a SB/A. The map f p- : 83/A a S3 is epi, carries the point p(A) to b in $3 and preserves complements. Bing shows in [4] that the suspension of $3/A is topologically 84. The suspension of p(A) is a wild arc W C284 and the suspension Of b is a tame arc T<: 84. Using the product structure Of the suSpension we can extend fp-1 to a map 4 F : S a SA. By construction F(W) = T and F(S4 - W) = S4 - T. 19 20 Theorem 4.2 For any k.> 0, there exists a wild k-cell WkC:Sk+3 which is weakly equivalent to a tame k-cell Tk<:.Sk+3. In fact Wk' goes homeomorphically onto Tk. Proof: We induct on k. Theorem 4.1 bases the induction. The product structure of the suspension gives the inductive step. k k+2 Theorem 4.3 For k >-0, there exists a wild k-cell W <: S which is weakly equivalent to a tame k-cell. Proof: We again induct with Corollary 2.8 used as a base for the induction. CHAPTER V 3 A MAP ON E PRESERVING OOMPLEMENTS a.e. In this chapter we weaken condition 2.) of Definition 3.1 so that the map need only preserve complements modulo a set of 3-dimensional Lebesgue measure zero. We use the concept of decomposition Space to construct the desired function for certain wild arcs. The concept of decomposition space comes from R. L. Moore, L. Vietoris, and P. Alexandroff. The following definition is fromTWhyburn [17]. Definition 5.1 Let X be a separable metric Space, a decomposition G of X is a representation = A A G X U a a a E where the Ad's are closed and disjoint. The decomposition ‘gpgpg. X' of G is formed by taking elements of G as points of X', and Open sets in X' are those sets Of elements of G whose union is Open in X. 3 In [5] Bing gives an example of decomposition of E whose resulting decomposition Space is again E3. As elements of G he chooses two linked circles go and g1 which bound discs perpendicular to each other and intersecting along a common radius. He also chooses a parameterized family of 21 22 figure-eight's, gt, 0 s t s l, where the loops of gt have radii t and l - t. The other elements of G are points of E3. Figure 5 Persinger [15] has exhibited a family of wild arcs in E which are equivalent to arcs embedded in a 3-book in E3. If we assume one of these arcs in a 3-book is embedded in the union of the discs in Bing's example, when the decomposition Space is Obtained we will have mapped a wild arc in E3. onto a segment in E3 via a map defined on all of E3. This map is seen to preserve the complement of the wild arc modulo a set of 3-dimensional Lebesgue measure zero. CHAPTER VI SOME POINT-SET RESULTS AND PATHOLOGY This chapter contains more definitions resulting from the study Of the Antoine problem. We use these new definitions to briefly develop some basic point-set properties. Definition 6.1 A function f: X a Y is IAi-piece continuous if there exists an index set A and a decomposition X ==UXa, a E A such that f restricted to Xa is continuous for each a. Example 6.2 Any compact metric Space X is the 2-piece con- tinuous image Of the closed unit interval 1. Let I = C U (I - C) where C is a cantor set. There is a con- tinuous surjection f1 : C d X. Define f2 : l - C d X to be a constant function. 3 Theorem 6.3 Let W be an arc in E and A a tame arc. 3 3 Then there exists a 2-piece continuous f: E a E such that 3 f1(W) =A and £2023 -w> =E -A. We may use the above definition to formulate the following question. Given a topological space X and an integer k, what can be said about X if every f :.X -¢Y is k-piece continuous? 23 24 Theorem 6.4 If every f : X a Y is 1-piece continuous, then X is discrete. Proof: Choose Y to be a discrete space with the same cardinality as X and let f : X ~‘Y be a bijection. Since f is l-piece continuous, the pre-images of open sets, hence the pre-images of points, are open. Thus points are open in X and X is discrete. Theorem 6.5 If every f : X a Y is 2-piece continuous, then X = XI U X2, X1 n X2 = ¢, X1 and X2 are discrete but X need not be. We may conclude however that every point x Of X has basic neighborhoods of the form x U U where x 6 Xi’ U<: xj, i # j. Proof: Again we choose Y discrete having the same cardinality as X and f : X a Y a bijection. Since f is 2-piece con- tinuous, X = X1 U X where f restricted to X1 is continuous. 2 Each xi must be discrete and we may choose them disjoint. X need not be discrete. If we choose Y to be a 2-point Space, each X1 is closed. The proof of Theorem 6.5 generalizes to n-piece con- tinuous. We may state this as follows: Theorem 6.6 If every f : X «‘Y is n-piece continuous, then X . X1 U X2 U°"U xn’ xi_n x, = ¢ if i # j, and each Xi J is discrete. 25 We also consider some examples in euclidean Space deal- ing with pathology in measure and dimension. We ask.what rela- tion, if any, exists between mi(A) and dim (A) where A is a compact set in B“. We also mention some known facts about what continuous functions and homeomorphisms do to measure and dimension. It is well known that continuous functions do not pre- serve either measure or dimension. A constant function can de- crease both measure and dimension, and space filling curves will increase both. It follows from the inductive definition Of dimension that homeomorphisms preserve dimension; but in the case of measure this is not true. Example 6.7 We give an example of a homeomorphism which does not preserve measure. The homeomorphism given will also extend to the whole Space. Let C be the usual cantor set in E1, C c:[0,l]. We know M1(C) = 0 = dim (C). But there is a homeomorphism h : C - C' where C' is a "fat" cantor set such that m1(C') = p, 0 < p < l, and dim (C') = 0. Since cantor sets are equivalently embedded in E1, the homeomorphism extends. This same idea can be used in higher dimensional euclidean Spaces. We next give an example of a wild set W in E3 having m3(W) = dim (W) = 0, yet the projection 1100 into the x-y plane has dim n(W) = l. 26 Example 6.8 Consider "Antoine's Necklace" a wild cantor set in E3. We shall construct Antoine's Necklace in such a way that the 3-dim measure of W is zero. Let the first solid torus T, in the construction have volume V. By considering a torus in T whose cross section is concentric with that of T and has a radiusilA/Ztines that Of T, we Obtain a torus whose volume is k V (by a theorem of Pappus). In this new torus we construct the 4 linked tori prescribed in the first stage of the construction of W. The total volume of these four tori is less than k V. We continue the construction by constructing in each Tij four linked tori whose total volume is less than k of the volume of Ti In the limit we will j' have a wild cantor set with the added property that the measure is zero. When we project W into the x-y plane we have an Object of dimension 1. At each Stage of the construction the projection is a continua. And stage by stage these continua are nested. Hence in the limit the intersection is a continua. Therefore since n(W) is connected and has more than one point, it has dimension at least one. If dim.n(W) = 2, then by Theorem IV.3 Hurewicz and Wallman, n(W) contains a disc. Say this disc has radius 3. If we look at the pre-image of this disc, it lies in one of the solid tori but cannot lie in all of them since their cross sections go to zero. Therefore dim n(W) = l. BIBLIOGRAPHY 10. ll. BIBLIOGRAPHY J.W. Alexander, "An example of a simply connected sur- face bounding a region which is not simply connected", Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 8-10. L. Antoine, ”Sur l'homeomorphisme de deux figures et de leurs voisinages", J. Math. Pures Appl. 4 (1921), 221-325. G.A. Atneosen, "On the Embeddability of Compacta in n-books: Intrinsic and Extrinsic PrOperties", Ph.D. Thesis, Michigan State University (1968). R.H. Bing, "The cartesian product of a certain non- manifold and a line in E4", Ann. of Math. 70 (1959), 399-412. R.H. Bing, "Decomposition of E3", Topolpgy of 3-manifolds and Related Topics, Prentice-Hall, Englewood Cliffs (1962). M. Brown, "A proof of the generalized Schoenflies theorem”, Bull. Amer. Math. Soc. 66 (1960), 74-76. P.H. 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