II "I II III I III! I I I I I I I I I II I HTHS CONTINUOUS NEAR HOMOGENEITY- Thesis for #he Degree» 0.; Ph. D. MECHIGAN STATE UNIVERSETY Hudson Van Ei'mn Kronk 1964- w'tun l n. LID. 11—15515 7 This is to certify that the thesis entitled Continuous Near-Homogeneity presented by Hudson Van Etten Kronk has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics d/m Q #2425“: , , Major professfr Date July 28, 1961+ 0.169 LIB RA R Y Michigan State University PLACE IN RETURN BOX to remove We checkout from your moord. TO AVOID FINES return on or batons due due. DATE DUE DATE DUE DATE DUE ‘- ‘Alrfifitll L__l ; I___ __ MSU Is An Affirmative ActiorVEqual OpportunIty Institution CWMS ABSTRACT CONTINUOUS NEAR-HOMOGENEITI by Hudson.Van Etten Kronk In a recent paper [1], P. Doyle and J. Hocking intro- duced the concept of continuous invertibility and inves- tigated its application to continua. The first part of this thesis deals with the analogous but weaker concept of continuous near-homogeneity. The object here being to generalize the results in [1] to continuously near- homogeneous spaces and also to study continuously near- homogeneous plane continua as a special case. Among the main results obtained are: (1) A compact set in En+1 is an n-sphere if it is con- tinuously near-homogeneous and contains an n—sphere. (2) Every decomposable continuously near-homogeneous plane continuum is a simple closed curve. (3) Every proper subcontinuum of a continuously near- homogeneous plane continuum is an arc. (a) Every continuously near-homogeneous plane continuum separates the plane. is a by-product of this part of the investigation, several prOperties of the continuous orbits in a continuously near- homogeneous indecomposable plane continuum are established. In particular, such orbits are identical with the composents Hudson Van.Etten Kronk of such a continuum and each such orbit is the image of the real line under a one-to-one continuous transformation. The second part of the thesis is concerned with the localization of continuous near-homogeneity. The principal result obtained is a characterization of those plane Peano continua which are continuously near-homogeneous at one or more points. REFERENCE 1. P.H. Doyle and J.G. Hooking, Continuously invertible spaces, Pacific J. Math.. Vol. 12 (1962) pp. “99-503. CONTINUOUS NEAR-HOMOGENEITY By Hudson Van Etten Kronk A THESIS Submitted to Hichigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1964 ACKNOWLEDGEMENT I wish to express my sincere appreciation.to Professor J.G. Booking for his help and encouragement throughout the course of this research. 11 To Ginny 111 TABLE OF CONTENTS Section I. Introduction e e e e e e e e Section II. Fundamental Definitions . . . . Section III. Continuously Near-Homogeneous Spaces Section IV. Continuously Near-Homogeneous Plane Continua . . . . . . . . Section V. Local Continuous Near-Homogeneity . Bibliography . iv 16 33 n “5 Figure 4.1 Figure 5.1 Figure 5.2 Figure 5.3 LIST OF FIGURES 16 35 36 b2 SECTION 1 INTRODUCTION In [10] P. Doyle and J. Hocking introduced the concept of continuous invertibility and investigated its applications to continua. The first part of this thesis deals with the analogous but weaker concept of continuous near-homogeneity which was introduced in [11]. The object here is to gen- eralize the results in [10] to the case of a continuously near-homogeneous space and also to study continuously near- homogeneous plane continua as a special case. Some of the principal results obtained are: Theorem 3.10: A compact set in En+1 is an n-sphere if it is continuously near-homogeneous and contains an n-sphere. Theorem 4.12: Every decomposable continuously near- homogeneous plane continuum is a simple closed curve. Theorem 4.1M: Every proper subcontinuum of a contin- uously near-homogeneous plane continuum is an arc. Theorem 4.18: Every continuously near-homogeneous plane continuum separates the plane. (Concerning this last result, it is an open question to determine precisely the number of complementary domains which a continuously near—homogeneous plane continuum can have.) As a by-product of this first part of the thesis, several properties of the continuous orbits in a contin- uously near-homogeneous indecomposable plane continuum are established. For instance, such orbits are identical with l the composants of such a continuum and each such orbit is an image of the real line under a one-to-one continuous transformation. The second part of this thesis investigates the con- cept of local continuous near-homogeneity which was intro- duced in [11]. In particular, the following characterization of plane Peano continua which are continuously near-homo- geneous at one or more points is obtained: Let K be a plane Peano continuum with non-empty CN(K). If K is one- dimensional, then K is the union of (at most) a countable number of simple closed curves having only one point p in common and all but a finite number of these simple closed curves have diameter less than any previously assigned positive number. Moreover, K is a simple closed curve if and only if CN(K) contains more than one point. If K is two~dimensional and CN(K) contains more than one point, then K is a closed disc. If CN(K)=p and if p is a non-cut point of K, then K is a closed singular disc and, finally, if CN(K)=p and p is a cut point of K, then K is a union of a countable number (22) of continua of the types already described. SECTION 2 FUNDAMENTAL DEFINITIONS In this section we present those definitions which are basic to continua theory and which are used in this thesis. A topological space S is said to be connected if the only two subsets of S that are simultaneously open and closed are S itself and the empty set C. A subset X of S is connected if it is connected with respect to the relative topology. Each point x of S belongs to a unique maximal connected subset of S called a component g£_;. The components of a space constitute a partition of the space into maximal connected closed subsets. If X is a closed proper subset of S, then every component of S-X(the com- plement of X in S) is called a complementapy_domain of X. If each two points of S can be joined by an app (the homeo- morphic image of the unit interval I=[o,1]) in s, then s is said to be arcwise connected. An arc component of S is a maximal arcwise connected subset of S. A compact connected set containing at least two points is called a continppm, There are two quite different types of continua, the decomposable and the indecomposable. A continuum is decomposable if it is the union of two proper subcontinua; otherwise it is indecomposable. If p is a point of a continuum M, then the union of all proper sub- oontinua of N that contain p is called a composant of M. u If a Hausdorff continuum is indecomposable, then its com- posants are equivalence classes and are uncountable in number. A continuum is indecomposable 23‘;ggggwg if it is the union of n continua such that no one of them is a sub- set of the union of the others and it is not the union of n+1 such continua. A.proper subcontinuum K of a continuum H is called a continuum.g§ condensation if every point of K is a limit point of H-K. It is easy to show that a Hausdorff continuum is indecomposable if and only if each of its proper subcontinua is a continuum of condensation. A space S is said to be locally connected ggflg,pgig§ p§§,if each open set U of p contains an open connected set 'V of p. A space S is said to be apcsygdetic 52 Egg if for each point th distinct from p, there exists a closed connected set H and an open subset U such that S-q‘DHDU‘Dp. A space is said to be locally connected (aposygdetic) if it is locally connected (aposyndetic) at each of its points. A continuum H is said to be hereditarily locally connected provided every subcontinuum of H is locally connected. A.1oca11y connected metric continuum is called a 23332 ccntinugg. .A fundamental result concerning Peano continua is the HathMazurkiewicz Theorem; which states that a nec- essary and sufficient condition that a space be a Peano continuum is that it be the image of the unit interval under a continuous mapping into a Hausdorff space. Every Peano continuum.is arcwise connected. A Peano continuum that doesn't contain a simple closed curve (the homeomorphic image of the unit circle) is called a dendrite. A 922 pgigg of a connected space S is a point péS such that S-p is disconnected; otherwise p is a non-cut pgggg of S. A point pES is said to be a Eggg_gg§'pg;g§ of S if there exists two points q,r€S-p such that every closed connected subset of S that contains both q and r also con- tains p. In general a weak cut point is not a cut point, however, the two concepts are equivalent for Peano continua. A point p of a connected space S is a lgggl‘gg§,pgip§ of S if it is a cut point of an open connected subset U of S. A space S is said to be cyclicly connected provided that every two points of 8 lie together on some simple closed curve in S. A Peano continuum is cyclicly connected if and only if it has no cut points (Cyclic Connectivity Theorem). A continuum H is said to be of M 1333 M 93 Eggs}; ‘gg‘g 35 pg! if for each open set V of p, there exists an open set U of p with UCV and such that the boundary of U contains at most n points of II. If H is of orderSn at pen, but not of orderSn-l, then M is said to be of o_r_de__r g g}; 2. A branch point of a continuum is a point of order greater than two. A space S is homogeneous if for each pair of its points p,q, there exists a homeomorphism of 3 onto itself which carries p into q. A space S is locally homogeneous if for each pair of its points p,q, there is a homeomorphism be- tween two open subsets of 8, one containing p, the other containing q, such that p is mapped into q. A space S is near-homogeneous (invertible) if for each point pES (proper closed set C of S) and each non-empty open set U of S there exists a homeomorphism of S onto itself which carries p (C) into U. SECTION 3 CONTINUOUSLY NEAR-HOMOGENEOUS SPACES This section deals with the general continuously near~ homogeneous space. We use the symbol C(S) to denote the group of all homeo- morphisms of the topological space S onto itself. Definition 3.1 An isotopy of a topological space S is a continuous map H:SXI-€>S with the properties: (1) If we define ht:s—+>s by setting ht(x)=H(x,t), then for all teI,hteG(S). (2) h0(x)=x for all xeS, i.e., ho is the identity mapping of 8 onto itself. Usually, we shall use ht to designate the isotopy. We remark that most authors omit property (2) in defin- ing an isotopy of S. Definition 3.2 A homeomorphism heG(S) is said to be a ggfpp- mation of S if there exists an isotopy ht of S with hl=h. The set of all deformations of S is denoted by H(S). It is easy to verify that H(S) forms a normal subgroup of 6(3). Definition 3.} The set of all images of a point ch under deformations of S is called the continuous orbit of x and is denoted by P(x). The continuous orbits of a space S are invariant under deformations of S, i.e., if heH(S), ch, then h(P(x))=P(x). Also, if x and y are any two points of S, then either P(x)= P(y) or P(x)f\P(y)=¢, i.e., the continuous orbits of S are equivalence classes. These properties follow directly from the fact that H(S) forms a group. Definition 3.“ The isotopy path of a subset XCIS under an isotopy H of S is the image of XII under H. Definition 3.5 A topological space S is continuouply Egg;- homogeneous if, for each point x€S and each non-empty open set U of 3, there is a deformation h6H(S) such that h(x)£U. Obviously, a continuously near-homogeneous space is near-homogeneous. The concepts of a continuously invertible space and a continuously homogeneous space are defined in an analogous manner. It is clear that a space S is con- tinuously near-homogeneous if and only if each of its con- tinuous orbits is dense in S and is continuously homogeneous if and only if it has exactly one continuous orbit. Before stating any results, we first mention some examples. Euclidean n-space, E”, is an example of a con- tinuously homogeneous space. For let, a=(a1,a2,...,an) and b=(b1,b2,...,bn) be any two points in En. Define H:EnII"’ En by setting H(xl,...,In,t)=(11+t(b1-31),...,Xn+t(bn-Cnl)- The map E is easily seen to be an isotopy of En and H(al,...,an,l)=(bl,b2,...,bn). More generally, it can be shown that any n-manifold (a separable connected metric space each of whose points has a neighborhood homeomorphic to E”) is continuously homogeneous. For example, if H is a l-manifold and a and b are any two points in M, then a suitable isotopy may be constructed as follows: Since M is a manifold, there exists in M a closed l-cell V contain- ing a and b in its interior. Let g be a homeomorphism from V onto [0,1] and assume without loss of generality that OEn+1 be a continuous map such that H|(XXt) is a homeomorphism of XXt into En+1, for all tsI. If H(XXI) doesn't contain either p or q, then p and q are separated by H(th), for all teI. Theorem 3.10 Let M be a compact continuously near-homo- geneous subspace of En+l. If M contains an n-sphere S, then M=S. £3922: Assume that M-S is not empty. Without any loss of generality, we may assume that there exists a point p of M in the bounded component A of En+l-S=ALJB. Let q be any point of S and U an open neighborhood of p such that Sf\U is empty. Since M is continuously near-homogeneous, there is an isotopy ht of M such that hl(Q)eU. Now consider the intersection V of A and the unbounded component of En+1-hl(S). Clearly v is not empty. Either V lies entirely in the isotopy path of S or there is a point x in V not covered by H(SXI). In the latter case, the point x and a point y in BfW(En+l-M) are separated by S, but they are not separated by h1(S). This contradicts Lemma 3.9 since the isotopy path of S doesn't contain either x or y. On the other hand, if V lies entirely in the iso- topy path of S, then V lies in M and hence M contains an open (n+l)-ce11. Thus by Theorem 3.8, M would be a compact (n+1)-manifold embedded in En+l. This is known to be impossible. Having been led to a contradiction in either 14 case, it follows that the point p is non-existent and hence M=S. Corollary 3.11 The only continuously near-homogeneous plane Peano continua are the simple closed curves. 2392;: By Corollary 3.7, such a space contains a simple closed curve (a l—sphere) and hence, by Theorem 3.10, is a simple closed curve. Corollary 3.12 Let M be a continuously near-homogeneous plane continuum other than a simple closed curve. Then every two points in a continuous orbit of M are joined by a unique arc in the orbit. Epggg: By Theorem 3.3, each two points in the same continuous orbit are joined by an arc. If there were another are joining them, then M would contain (and hence be) a simple closed curve. Corollary 3.11 is similar to Mazurkiewicz's result [17] that every homogeneous plane Peano continuum is a simple closed curve. This result was later generalized by H.J. Cohen [7]. who showed that one need only require the homogeneous continuum to contain a simple closed curve. Cohen's result is thus similar to a special case of our Theorem 3.10. In connection with these remarks, it should be noted that there do exist near-homogeneous plane con- tinua other than simple closed curves. A well known example of such a continuum is the universal plane curve (the con- tinuum obtained by considering a square and successively 15 deleting first the (open) middle-ninth of that square, second the middle-ninths of each of the eight squares remaining, third the middle-ninths of each of the sixty four squares remaining, etc.). A proof of the near- homogeneity of the universal plane curve is given in [9]. C.E. Burgess has shown [6] that, if a metric continuum is homogeneous and hereditarily locally connected, then it is a simple closed curve. The last theorem of this section shows that the corresponding theorem for continuously near- homogeneous, hereditarily locally connected metric continua is also true. Theorem 3.13 If the metric continuum M is continuously near- homogeneous and hereditarily locally connected, then M is a simple closed curve. 2292;: Every hereditarily locally connected metric con- tinuum is separated by some countable set [20,p. 99]. Any countable set that separates M contains a local cut point x of M [20, Corollary 9.41, p. 62]. Hence, M contains a continuous orbit P(x) of local cut points and since all but a countable number of the local cut points of any metric continuum are points of order two [20, Theorem 9.2, p. 61], every point of P(x) is of order two. This, however, implies that P(x) is the only continuous orbit of M and thus every point of M is of order two. K. Menger has shown [18] that a simple closed curve is the only metric continuum, each of whose points is of order two. SECTION 4 CONTINUOUSLY NEAR-HOMOGENEOUS PLANE CONTINUA In [4], R.H. Bing describes an indecomposable plane continuum K, each of whose proper subcontinua is an are, such that K is near-homogeneous but not homogeneous. This continuum is pictured in Figure 4.1. A B ‘ C .__x-qx05 Figure 4.1 The example K intersects the x-axis in a (topological) Cantor set and is the union of semicircles with ends on this Cantor set. The continuum K is obtained by starting with a punctured disc D with three holes and digging canals (in the manner described on pages 222, 223 of [4]) into the disc from the four complementary domains of D. The continuum l6 17 K has the properties that each arc component is dense in K and each arc in K lies in an open subset homeomorphic with the Cartesian product of the Cantor set and an open interval. 0n the basis of these two properties, Bing noted that K is near-homogeneous. To see that these two proper- ties also imply that K is continuously near-homogeneous, we proceed as follows: We first observe that if (a,bl), (a,b2)cCXI, where C denotes the usual Cantor set, then there is an isotopy ht of CXI such that (l) hl(a,bl)=(a.b2) (2) ht(x,0)=(x,0), ht(x,l)=(x,l) for all teI. (ht can be defined in the same manner as we defined the isotopy h of [0,1] on page 9) Now let p be any point in t K and let U be any non-empty open subset of K. Since the arc component of p is dense in K, there exists an arc pq in K such that qu. Let V be a open neighborhood of this are homeomorphic to the Cartesian product of the Cantor set and an open interval. Clearly V contains a closed set W homeomorphic to CXI and such that p and q are interior points of W. Let g:W-—)CXI denote this homeomorphism. As mentioned above, there exists an isotopy ht of CXI such that (1) hl(s(p))=s(q) (2) ht(X,O)=(x,0),ht(x,l)=(x,l) for all tel. Finally, the desired isotopy f of K is obtained by setting t 18 (l) ft(x)=g‘lhtg(x) for all er and (2) ft(x)=x for all xeK-W,teI. It would be nice to know if the continuum K is also con- tinuously invertible, but this appears to be a difficult open question. Motivated by this example, we proceed now to inves- tigate continuously near-homogeneous plane continua. We recall from Section 3 that the only such Peano continua are the simple closed curves. We shall eventually prove the much stronger result that the only decomposable continuously near-homogeneous plane continua are the simple closed curves. R.H. Bing has shown [4, Theorem 1] that every homo- geneous plane continuum that contains an arc is a simple closed curve. Since a continuously near-homogeneous plane continuum obviously contains an arc, we have the following result. Theorem 4.1 If M is a homogeneous, continuously near—homo- geneous plane continuum, then M is a simple closed curve. We next show that no continuously near~homogeneous plane continuum can contain a simple triod. (A simple triod is a continuum formed by three arcs px, py, and pz such that each pair of these arcs have just the point p in common; the point p is called the emanatipp_ppint of the triod.) Our result is similar to Cohen's result [7, Corollary 2.12] that no locally homogeneous plane continuum can contain a simple triod. There do exist, however, near-homogeneous 19 plane continua containing simple triods, e.g., the universal plane curve. Theorem 4.2 No continuously near-homogeneous plane continuum M contains a simple triod. Epppi: Assume that M did contain a simple triod T with emanation pointfz Since a triod is obviously not near- homogeneous, M-T is not empty and is an open subset of M. Let ht be an isotopy of M such that hl(P)eM-T. It follows from the Hahn-Mazurkiewicz Theorem that the isotopy path of T is a Peano continuum. If the isotopy path of T con- tained a simple closed curve, then by Theorem 3.10, M would be a simple closed curve contradicting our assumption that M contains a simple triod. Therefore, the isotopy path of T is a dendrite. But the isotopy path of the emanation point p contains an uncountable number of branch points in the isotopy path of T. This is impossible since no dendrite contains more than a countable number of branch points [20, Theorem 1.2, p. 89]. This theorem has several immediate corollaries. The first is a generalization of Corollary 3.11 and is similar to Cohen's result [7, Theorem 3] that every homogeneous arcwise connected plane continuum is a simple closed curve. Corollary 4.3. Every arcwise connected, continuously near- homogeneous plane continuum M is a simple closed curve. 2322:: We observe that M cannot have two continuous orbits. For if it did, there would exist an are joining 20 these orbits. Recalling that each point in a continuous orbit is an interior point of an arc in the orbit, it follows that M would contain a simple triod contradicting Theorem 4.2. Thus M contains just one continuous orbit (and hence is homogeneous) and Theorem 4.1 applies. Corollary_4.4 The arc componants and continuous orbits of a continuously near-homogeneous plane continuum M are the same. £3291: Let x be any point of M and let A denote the arc component of M containing x. It follows from Theorem 3.3, that P(x) is contained in A. If there were a point p in A-P(x), then by definition of A, there exists an arc p>xa in M. This, however, implies that M contains a simple triod. Therefore A-P(x) is empty and thus A:P(x). Corgllary_4p5 If M is a continuously near-homogeneous plane continuum, then every proper Peano subcontinuum of M is an arc. £3992: Let N be a proper Peano subcontinuum of M. By Theorem 3.10, N doesn't contain a simple closed curve. Thus N is a dendrite and every dendrite other than an arc contains a simple triod. We now prove the very useful result that no proper subcontinuum of a continuously near-homogeneous plane continuum separates the plane. In [10], Doyle and Hocking used the corresponding theorem for continuously invertible plane continua to show that every proper subcontinuum of a continuously invertible plane continuum is an arc; from 21 which it followed that every continuously invertible plane continuum other than a simple closed curve is indecomposable. We shall eventually prove these same results for continuously near-homogeneous plane continua, but in the opposite order. Their proof that every sub- continuum of a continuously invertible plane continuum is an arc makes strong use of invertibility, which we aren't assuming. In proving Theorem 4.6, we again make use of Lemma 3.9 for E2 and the proof is analogous to the proof of Theorem 3.10. Theorem 4.6 No proper subcontinuum of a continuously near- homogeneous plane continuum separates the plane. £3923: Assume that X is a proper subcontinuum of M that separates the plane. Without loss of generality, we may assume that there exists a point p of M in a bounded complementary domain B of X. Let q be any point of X and let ht be an isotopy of M such that hl(q)eU, where U is an open neighborhood of p such that UT\X;¢. Since h1(X) is homeomorphic to X, hl(X) also separates the plane. Consider the intersection V of B and the unbounded complementary domain of hl(X). Clearly V is not empty and, as in Theorem 3.10, there are two cases to consider. Either V lies entirely in the isotopy path of X or there is a point x in V not covered by the path. In the first case, M would contain an open 2-cell which is clearly impossible. The 22 second case is also impossible, since then x and any point y in the unbounded complementary domain of X and not in the isotopy path of X would be separated by X, but not by hl(X). This contradicts Lemma 3.9. Hence the only sub- continuum of M that can separate the plane is M itself. Corollary 4.7 Every continuously near-homogeneous plane continuum M is the common boundary of each of its complemen- tary domains. 2329;: Since M is a nowhere dense subset of the plane, this is clearly true if M doesn't separate the plane. (Later we will show that this case can't occur.) Assume then that M separates the plane and let B be the boundary of a complementary domain of M. It is shown in [19] that B is a subcontinuum of M and since it is the boundary of a domain, it separates the plane. By Theorem 4.6, this is only possible if B=M. In connection with Corollary 4.7, we remark that the universal plane curve is near-homogeneous and not the boundary of each of its complementary domains. However, it is true that every homogeneous plane continuum is the boundary of each of its complementary domains [6, Theorem 2]. K. Kuratowski has shown [16, Theorem 3] that every plane continuum which is the common boundary of three or more complementary domains is indecomposable or is indecom- posable of index 2. Burgess has shown [5, Theorem 2] that the latter type can't be near-homogeneous. The following 23 result, which will later be generalized, is then immediate. Theorem 4.8 Every continuously near—homogeneous plane continuum which has three or more complementary domains is indecomposable. Definition 4.1 Let M be a continuously near-homogeneous plane continuum other than a simple closed curve and let p and q be two distinct points in the same continuous orbit of M. The union of all the arcs in M that have p as an end point and contain q is called a ray starting g3 p. A ray in the above sense differs from an ordinary ray of the plane in that it is neither straight nor closed. However, it has a starting point and as a consequence of the next lemma is the image of an ordinary ray under a one- to-one continuous transformation. Lemma 419» Let M be a continuously near-homogeneous plane continuum other than a simple closed curve and let H be a ray in M with starting point p. Then R is the union of a countable number of arcs ppl, pp2, pp3,... such that (l) for each i and j either ppiQppJ or ppJCppi and (2) each arc ppi is a proper subset of some ppj. 2399:: Let {pisbe a countable dense subset of R. Then R is the union of the arcs ppl, pp2, pp3,..., since if there were a point r of R not in any ppi, then consider the arc pr. By Theorem 3.3, the arc pr may be extended to an arc ps such that r is contained in the interior of ps. Since M contains no simple triod, each pi belongs to the 24 are pr. But then {piSwould not be dense in R, since no pi is near 8. That the sequence of arcs {ppi33atisfies (1) follows directly from Corollary 3.12. Property (2) follows from the denseness of the pi. Lemma 4.10 Let M be a continuously near—homogeneous plane continuum other than a simple closed curve. Then for each point x of M, P(x) is the union of two rays L,R, starting at x such that Lr‘st. Hence P(x) is the image of El under a one-to-one continuous transformation. Egggg: By Theorem 3.3, x is an interior point of an arc ab in P(x). It follows from the fact that M contains no simple triods that P(x) is the union of two rays starting at x and going through a,b respectively. Since M contains no simple closed curve, these rays intersect only at x. Since P(x) is the one-to-one continuous image of El, its points can be simply ordered in an obvious way. If P(x)=LLJR, where L and R are rays starting at p and passing through a and b respectively, then we will always order P(x) such that ahl(b), then for some value of g,0r2>r3>"'and having r as a limit point. Now let h be an isotopy of M such t that hl(r)=u, where u is any point in Lf\(M?H). Since hl is a homeomorphism of M, the sequence of points hl(rl), h1(r2), hl(r3),... should have u as a limit point. By Lemma 4.11, the isotopy ht preserves the order of points so that hl(r)=u>h1(s)2h1(r1)>hl(r2)>"°. This, however, implies that the set of points hl(rl), hl(r2),... can't have u as a limit point unless the ray R2 returns to L which is contrary to our assumption. Having been led to a contradiction in both cases, it follows that M must be indecomposable. F.B. Jones has shown [13] that a homogeneous plane continuum is a simple closed curve if it is either apo- syndetic at some point or contains a non-weak cut point. Since an indecomposable continuum is not aposyndetic at any of its points [14] and consists entirely of weak out points, the analogous result for continuously near-homo- geneous plane continua follows directly from Theorem 4.12. Corollary 4gl3. A continuously near-homogeneous plane con- tinuum is a simple closed curve if it is either aposyndetic at some point or contains a non-weak cut point. We now use Theorem 4.12 to show that every proper sub- 27 continuum of a continuously near-homogeneous plane con- tinuum is an arc; thus generalizing Corollary 4.5. Theorem 4.14 Every proper subcontinuum of a continuously near-homogeneous plane continuum M is an arc. 2239;: Clearly this is true if M is a simple closed curve. If M is other than a simple closed curve, then by Theorem 4.12 M is indecomposable and hence every proper subcontinuum of M is a continuum of condensation. Assume that K is a proper subcontinuum of M other than an arc. Let x be a point of K that is a limit point of points in K not in the continuous orbit of x. Such a point exists, since K is not an arc. Let ht be an isotopy of M such that hl(x)=x' is a point of the Open set M-K. It follows from the continuity of hl and the choice of x that there exists a point y in K-P(x) such that hl(y)=y' also belongs to M-K. Let L be an irreducible subcontinuum of M between x and y [12, Theorem 2-10] and let hl(L)=L'. This subcontinuum L is unique and is contained in K. To see this, assume that N is another irreducible subcontinuum of M between x and y. Then LfNN must be connected, since otherwise LLJN would separate the plane (this follows from the well known Janiszewski-Mullikan Theorem [19, Theorem 22, p. 175]). But by Theorem 4.6,this would be possible only if LLJN=M. This in turn would imply that M is decomposable, which is contrary to our assumption. Consider now the proper sub- continuum of M formed by L,L' and the arcs xx' and yy'. 28 This subcontinuum is the union of two subcontinua B and D, where B consists of L, xx', and yy' and D consists of L', xx', and yy'. Now Bf\D must be disconnected, since other- wise it would be a proper subcontinuum of L containing x and y, which contradicts the fact that L was chosen to be irreducible between these two points. But if BFWD is dis- connected, then by the Janiszewski-Mullikan Theorem B\JD would separate the plane. As we have previously observed, this is not possible by Theorem 4.6 and the fact that M is indecomposable. Having arrived at a contradiction, we can conclude that every proper subcontinuum of M is an arc. Corollary 4.15 Let M be a continuously near-homogeneous, indecomposable plane continuum. Then the composants of M and the continuous orbits of M are identical. 2399;: Let x be any point of M and let C be the com- posant of M containing x. By definition C is the union of all proper subcontinua of M containing x and hence P(x) is a subset of C. If C were not a subset of P(x), then there would exist a proper subcontinuum of M joining x to a point in C-P(x). But by Theorem 4.14, this subcontinuum is an arc and hence M would contain a simple triod which is impossible. We conclude this section by showing that every con- tinuously near-homogeneous plane continuum has at least two complementary domains. Before proving this, however, we need a couple of lemmas. 29 Lemma 4.16 Let a and b be any two points in the same con- tinuous orbit of an indecomposable, continuously near-homo- geneous plane continuum M. Then there exists a continuous map HzMXI-—)M with the properties: (1) If ht:M-+;M is defined by ht(x)=H(x,t), then, for all th, hteG(M) (2) ho(a%aa,kh}eizb (3) H(ex1)=eb. 3399;: Let G be an isotopy of M such that g1(a)=b. Clearly it is sufficient to show that there is a subinterval [tl,t2] of I such that gtl(a)=a,gt2(a)=b, and G(ax[tl,t2])= ab. Assume that the continuous orbit of a is ordered such that aM such that hO(y)=y, hl(y)=z, and H(yxl)=yz. Then hO(albl), hQ(a2b2),... is a folded sequence of arcs converging to the arc ho(xy)= hO(x)y such that h0(al),ho(bl),h0(a2),h0(b2),... converges to ho(x). If P(x) is ordered such that x diameter M/4-—l/i and both ends of xy lie in the same link 0‘ of D1. If diameter xyif dia- meter M/2, then we reduce this arc by throwing away the part of it inO< and consider one of the larger components. of the remainder. These components are arcs and at least one of them, say x'y', must have diameter greater than dia- meter M/4- l/i. For otherwise, it would follow that dia- meter xy_<_l/i+2(diameter M/4-l/i)=diameter M/2- l/i. The end points of x'y' lie in the same linkB of Di’ since if 32 they didn't M wouldn't be tree—like. If the arc x'y' doesn't suffice, then we reduce it by throwing away the part of it not in and consider one of the larger com- ponents of the remainder, etc. Eventually we shall either obtain an arc of the desired diameter or one having a sub- arc of the desired diameter. The sequence of arcs {albig has a convergent subsequence [20, Theorem 7.1, p. 11] and the limit of this subsequence is a closed, connected sub- set of M [20, Theorem 9.11, p. 15]. Clearly the limit con- tains at least two points and thus is a proper subcontinuum of M. By Theorem 4.14, this subcontinuum is an arc ab. However, there is a folded sequence of arcs (each in one of the albi's) converging to a subarc of ab. This contradicts Lemma 4.17. Thus a continuously near-homogeneous plane continuum must have at least two complementary domains. The exact number of complementary domains such a continuum can have remains an open question. SECTION 5 LOCAL CONTINUOUS NEAR-HOMOGENEITY In this section we consider the concept of a space being continuously near-homogeneous at a point and apply this concept to plane Peano continua. Definition 5.1 A topological space S is said to be 922: tinuously near-homogeneous at 5 221g; 2 if for each point q€S and each open neighborhood U of p, there exists a homeomorphism h€H(S) such that h(q)éU. If we only require h to be in 6(3), then S is said to be near-homogeneous at p. The set of points at which 3 is continuously near-homo- geneous (near-homogeneous) is denoted by CN(S) (N(S)). Evidently, a space S is continuously near-homogeneous if and only if CN(S)=S and near homogeneous if and only if N(S)=S. The first few preliminary results of this section appeared in [11]. We give proofs here for the sake of com- pleteness. For the first five results of this section on CN(S), the corresponding result for N(S) (using 6(8) in place of H(S)) is also true and is proved in an analogous manner. Theorem 5.1 For any space S, the set CN(S) is carried onto itself by each h€H(S). 2322;: Let p be any point of CN(S) and let h£H(S). we want to show that h(p)€CN(S). Let U be any open neigh- borhood of h(p) and let q be any point in S. Then h'1(U) is an open neighborhood of p and hence there exists a de- formation gems) such that g(h'1(q))€h'1(U). Then hgh'1(q )QU 33 34 and hgh-leH(S). Hence, by definition h(p)eCN(S). Theorem 5.2 The set CN(S) is a closed subset of S. 2392:: Let p be a limit point of CN(S) and let U be any open neighborhood of p. By definition of limit point, there is a point q¥p in CN(S)/WU. Thus for any point xeS, there exists a deformation h of S such that h(x)cU. It follows that peCN(s) and hence CN(S) is closed. Theorem 5.3 The set CN(S) is a continuously near-homo— geneous subspace of S. Eggggz Let p and q be any two points in CN(S) and let V be an open neighborhood of p in the subspace topology of CN(S). By definition of the subspace topology, there is an open neighborhood U of p (open in S) such that V=Uf\CN(S). Let h be a deformation of s such that h(q)eU. By Theorem 5.1, h[CN(S)]=CN(S) and hence h|CN(S) is an element of H(CN(S)) which carries the point q into UFNCN(S)=V. Thus CN(S) is continuously near-homogeneous. In order to have an example or two on hand, we note that, for the closed n-cube In,n>l, CN(In) is a topological (n-l)-sphere. An example in which CN(S) consists of exactly one point is pictured in Figure 5.1. It consists of the tangent circles x2+(y-l/2n)2=l/22n,n=l,2,...,. We will later show that every one-dimensional plane Peano continuum having exactly one point in CN(S) may be homeomorphically embedded in this continuum. We remark that such Peano con- tinua are often called roses. 35 Figure 5.1 Theorem 5.4 If CN(S) contains a non-empty open subset of S, then S is continuously near-homogeneous. 3399;: Suppose that CN(S) contains an open set U of S. Since U is an open neighborhood of a point in CN(S), each point ch can be carried into U by a deformation h of S. This implies that h(x) (and hence x) is in CN(S). There- fore CN(S)=S. Thus for any space S, the set CN(S) is either all of S or is a closed, nowhere dense subset of S. The next re- sult is of interest in examples where CN(S) is a proper sub- set of S containing more than one point. Theorem 5L5 Let S be a space with non—empty CN(S). Then CN(Q) is also non—empty, where Q denotes the quotient space S/CN(S). Proof: Let p:S-—)Q be the natural projection map and let p(CN(S))=w. We show that w belongsto CN(Q). Let U be 36 an open neighborhood of w in Q and let q be any point of Q. If q=w, the identity map of CN(Q) carries q into U. If q#w, then p-1(q)=qu. By definition of the quotient topology, p-1(U) is an open neighborhood of CN(S) in S. Hence there exists heH(S) such that h(q)ep-1(U). Since CN(S) is invariant under elements of H(S), the composition php-1 is a one-to-one transformation of Q onto itself and is a homeomorphism because php-1 and (php-l)-1=ph-1p-l are both closed. Then php-leH(Q) and carries q into U. Hence weCN(Q). Theorem 5.5 gives rise to an interesting unsolved ques- tion. For each positive interger n>l, let Sn denote the quotient space Sn_l/CN(Sn_l) and let Sl=S/CN(S). Does there always exist an interger N such that for all nZN, the spaces Sn are all homeomorphic? An example for which N=3 is pic- tured in Figure 5.2. For this example, CN(S)=Sl and S/CN(S) =D, a (topological) 2-disc. Hence D/CN(D)=sl and Sl/CN(Sl) is a single point. Figure 5.2 37 Theorem 5.6 If S is a Hausdorff space, then CN(S) has 0,1 or an uncountable number of points. Egggg: Suppose that there are two points p and q in CN(S). Let U be an open neighborhood of p not containing q. The isotopy path of q under an isotopy of S carrying q into U is a non—degenerate continuous image of the unit interval in a Hausdorff space and hence is a Peano continuum. Every non-degenerate Peano continuum contains uncountably many points. Finally, it is evident that each point in this isotopy path is in CN(S). In analogy to Theorem 3.4, we have the following two results. Theorem 5,? If CN(S) is non-empty, then S is connected. £3993: Let peCN(S). Since the continuous orbit P(x) of each point xeS contains a point in every open neighbor- hood of p, it follows that peFTET. Therefore S=LJFT§7 is a union of connected sets, each containing the point p, and hence is connected. Theorem 5.8 For each space S, the set CN(S) is connected. Proof: This follows immediately from Theorems 5.3 and 5.7. The next theorem is a slight generalization of Theorem 8 of [11]. Theorem 5.9 Let S be a T -continuum with non-empty CN(S). 1 Then if S has a cut point p, it is the only cut point of S and CN(S)=p. 38 2322:: We show that CN(S)=p, by showing that each point qu-p is not in CN(S). Assume that q¥p is in CN(S). Since p is a cut point of S, S-p=ULJV, where U and V are disjoint, non-empty open sets in 3. Assume without loss of generality that qu. For each point er, there is an isotopy ht of S such that h1(x)eU. But then there must be some tO,O0. 23922: As we noted in the proof of Theorem 5.10, p and each point q¥p in K lie together on a simple closed curve in K. Since CN(K)=p, it follows that K contains at least two simple closed curves. Moreover, all simple closed curves in K contain the point p. For otherwise, we could apply Lemma 3.9 and the type of argument used in Theorem 3.10 to arrive at a contradiction of the fact that K is one-dimensional. Now let R and S be any two simple closed curves in K. We want to show that Rf\S=p. Assume that there did exist another point qufWS. Without loss of generality, we can assume that q is the emanation point of a simple triod T in K. Let U be an open neighborhood of p not containing q and let h be an isotopy of K such that hl(q)eU. Since t the point p remains fixed throughout the isotopy, the iso- topy path of T is a Peano continuum in K which doesn't contain 40 p. It follows that the isotopy path of T is a dendrite, since otherwise K would contain a simple closed curve not containing p. But the isotopy path of T contains an un- countable number of branch points and this is known to be impossible. Hence K is the union of simple closed curves, each two of which have only p in common. If S is any simple closed curve in K, then S-p is a component of K—p. Since an open subset of a Peano continuum contains only a count- able number of components, it follows that K consists of a countable number of simple closed curves. By Theorem 3.9 of [12], only a finite number of the simple closed curves in K have diameter greater than any positive number e>0. As an immediate corollary to the proof of Theorem 5.11, we get a result analogous to Theorem 4.2. Corollaryi5.12 Let M be a one—dimensional plane continuum with CN(M)=p. Then M-p contains no simple triod. Theorem 5.13 Let K be a plane one-dimensional Peano con- tinuum such that CN(K) contains at least two points. Then K is a simple closed curve. 3329;: Since CN(K) is closed and connected, it is a subcontinuum of K. By Theorem 5.10, K contains a simple closed curve S. Now CN(K) is a subset of S, for if there existed a point peCN(K)-S, we could apply Lemma 3.9 to show that K contained an open 2-disc contrary to the assumption that K is one-dimensional. Since CN(K) is continuously near- homogeneous and is a continuum, it follows that CN(K)=S. 41 Clearly S is the only simple closed curve in K and thus Kss. P. Alexandroff has shown [1] that the isotopy path of a simple closed curve S in E3 is at least two-dimensional provided that, under the isotopy h hl(S)#hO(S)=S. This t! result can be used to extend Theorems 5.11 and 5.13 to one- dimensional Peano continua in E3. The proofs would be iden- tical, except for the using of Alexandroff's result in place of Lemma 3.9. Theorem_5.l4 Let K be a two-dimensional plane Peano con- tinuum such that CN(K) contains at least two points. Then K is a closed 2-disc. nggf: Since CN(K) contains at least two points, it follows from Theorem 5.9 that K has no out points. Hence by [19, Theorem 46, p. 199], the boundary of each complemen- tary domain of K is a simple closed curve. Let S be the simple closed curve which is the boundary of the unbounded complementary domain. We show that CN(K)C:S and hence that CN(K)=S. Assume there did exist a point peCN(K)-S and let qu. We can carry q by a deformation h of S into an open neighborhood U of p, where U(\S=¢. The isotopy path of S under this isotopy would have to be two-dimensional by Lemma 3.9. This is a contradiction, since we would be mapping boundary points into interior points. Thus CN(K)=S and hence K has only one complementary domain. Being two-dimen- sional, it follows that K is a closed 2-disc. Suppose now that K is a plane Peano continuum with CN(K) 42 2p, where p is a non-cut point of K. It follows from Theorem 5.9 that K has no out point. As noted in the proof of Theorem 5.14, this implies that the boundary of each com- plementary domain of K is a simple closed curve containing p. we denote the union of these simple closed curves by L. Evidently, L contains at least two simple closed curves and each two have only p in common. If 86L is the boundary of any bounded complementary domain of K, then since p is a non-cut point of m, there are no points of K in the bounded component of Ez-S. Since p is a non-cut point of K, K does contain each point in the intersection of the bounded com- ponent of the unbounded complementary domain with the un- bounded compenents of the bounded complementary domains. If we call this set of points x, then KsLLJM. We shall call a continuum of this type a closed singular $332. The “pinched annulus”, Figure 5.3, is the simplest example of a closed singular disc. 43 Theorem 5.15, Let K be a plane Peano continuum with CN(K) =p, where p is a non-cut point of K. Then K is a closed singular disc. Before completing the classification of plane Peano continua K with non-empty CN(K), we state three lemmas. Lemma 5.16 Let K be a Tl-continuum with CN(K)=p, where p is a cut point of K. If C is a component of K-p, then ijp is carried onto itself under isotopies of K. £2991: Let ht be an isotopy of K. Since p is the only cut point of K, it remains fixed under ht' Now let x be any point of C. Since p is the only cut point of K, the isotopy path of x under ht lies in K-p. But the iso- topy path of x is connected and since C is a component of K-p, the path must lie in C. Therefore ht(x)eC, for all teI. Lemma 5.17 Let K be as in Lemma 5.16. Then for each com— ponent C of K-p, peCN(CLJp). nggf; Let V be an open neighborhood of p in ijp and let xeCk/p. Then V=Uf\(C\Jp), where U is an open sub- set of K. There exists an isotopy H of K such that h1(x)cU. By Lemma 5.16, H|(C\Jp) is an isotopy of ijp and hl|(CLJp) carries x into V. Lemma 5.18 Let K be as in Lemma 5.16. If C is a component of K-p, then CKJp has no out points. 2392;: By Lemma 5.17, peCN(C\Jp) and hence the only possible cut point of CLJp is p itself. Since (CLJp)-P=C. 44 p is a non-cut point of C p. Theorem 5.19 Let K be a plane Peano continuum with CN(K) =p, where p is a cut point of K. Then K is the union of a countable number (22) of continua of the types character- ized in Theorems 5.13, 5.14, and 5.15, each two of which have only the point p in common. Egggf: Let C be a component of K—p. Then Cka is a Peano subcontinuum of K. By Lemmas 5.17 and 5.18, peCN(CLJp) and Cka has no out points. Since we have characterized such continua previously in Theorems 5.13, 5.14, and 5.15, it follows that K is characterized. 1. 2. 3. 9. 10. 11. 13. 14. BIBLIOGRAPHY P. Alexandroff, Ueber endlich-hoch zusammenhangende stetige Kurven, Fund. Math., Vol. 13 (1924) pp. 34-41 T.C. Benton, 0n continuous curves which are homogeneous except for a finite number of points, Fund. Math., Vol. 13 (1924) pp. 151-177 R.H. Bing, Snake-like continua, Duke Math. J., Vol. 18 (1951) pp. 653-663 , A simple closed curve is the only homo- geneous bounded plane continuum that contains an arc, C‘nade J. Math., v01. 12 (1960) pp. 209-230 C.E. Burgess, Some theorems on n-homogeneous continua, Proc. Amer. Math. Soc., Vol. 5 (1954) pp. 136-143 , Continua and various types of homogeneity, Vol. 88, (1958) pp. 366-37“ H.J. Cohen, Some results concerning homogeneous plane continua, Duke Math. J., Vol. 18 (1951) pp. 467-474 P.H. Doyle and J.G. Hocking, A characterization of Euclidean n-space, Mich. Math. J., Vol. 7 (1960) pp. 199-200 , Invertible spaces, Amer. Math. Monthly, VBIT‘EE—T1531) pp. 959-965 , Continuously invertible spaces, Pacific J. __hath.,"_Vo1. 12 (1962) pp. 499-503 H.D. Guay, J.G. Hocking and H.V. Kronk, Local near- homogeneity, Amer. Math. Monthly, Vol 70 (1963) Ppe 827-833 J.G. Hocking and 6.8. Young, Topology, Reading, Mass.: Addison-Wesley, 1961 F.B. Jones, A note on homogeneous plane continua, Bull. Amer. Math. Soc., Vol. 55 (1949) pp. 113-114 , Concerning aposyndetic and non-aposyndetic continua, Bull. Amer. Math. Soc., Vol. 58 (1952), pp. 137-151 45 46 15. K. Kuratowski, Topologie, Vol. 2, 2nd ed. Warsaw: 1948 16. , Sur la struture des frontiére communes a'deux regions, Fund. Math., Vol. 12 (1928) pp. 20-42 17. S. Mazurkiewicz, Sur 1es continue homogenes, Fund. Math., Vol. 5 (1924) pp. 137-146 18. K. Menger, Kurventheorie, Teubner, Berlin-Leipzig, 1932 19. B.L. Moore, Foundations 0f Point Set Theory, Amer. Math. Soc. Colloquium.Publications, Vol. 13 (1962) 20. G.T. Whyburn, Analytic Topology, Amer. Math. Soc. Colloquium Publications, Vol. 28 (1942) (I Airs- ' MATH U3. 1"ng 1 6 "“5 W Dad-813'" Jim 7 23:53 i‘. in; ( 13.7., i. L. HY; "'1711113111111]!(1111111111111ES