IIIIII.) TOPOLOGICAL PROPERTIES OF ' COMPACTIFICATIONS OF A HALF ~0PEN INTERVAL Thesis for the Degree of Ph; De MICHIGAN STATE UNIVERSITY DAVID PARHAM BELLAMY 1968 L .IB R A R Y Michigan State University .mm TH ES: 5 This is to certifg that the thesis entitled TOPOLOGICAL PROPERTIES OF COMPACTIFICATIONS OF A HALF-OPEN INTERVAL presented by DAVID PARHAM BELLAMY has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics I M professor 0-169 ABSTRACT TOPOLOGICAL PROPERTIES OF COMPACTIFICATIONS OF A HALF-OPEN INTERVAL by David Parham Bellamy We make the following definitions: Let A = [1,”). A Hausdorff compactification (S,h) of A is a pseudocone, where h: A H S is the natural embedding. S-h(A) is the base of S; h(l) is the vertex of S; and if the base of S is homemorphic to a topological space X, S is a pseudocone 2333 g, Unless explicitly stated otherwise, both A and the base of a pseudocone S will be considered identified with their images in So When it is convenient to specify that X is the base of a pseudocone, the pseudocone will be denoted by P(X)g This is not meant to imply any sort of functorial relationship“ The broad questions considered are: ”Under what conditions does there exist a pseudocone over X?" and ”What is the relation— ship between the properties of P(X) and those of X?” An outline of principal results follows" The numbering of results below does not correspond to the numbering in the thesis. Chapter Ia Existence and general properties Theorem 1: Let P(X) be a pseudocone: Then P(X) is a continuum irreducible between its vertex and any point on X0 Theorem 2: The base of a psuedocone is a compact Hausdorff David Parham Bellamy continuum. Theorem 3: Let X be a compact Hausdorff continuum which is irreducible about some separable subset of itself and such that there exists a connected, locally path connected, locally compact Hausdorff space Y and an embedding f: X a Y such that f(X) is a G6 subset of Y° Then there exists a pseudocone over X0 Corollary 1: If X is a compact metric continuum, there exists a pseudocone over X0 Corollary 2: If X is a compact Hausdorff continuum with a separable dense path component, there exists a pseudocone over X, Chapter 110 Retracts of pseudocones Theorem: Let X be a compact Hausdorff space“ The follow- ing are equivalent: 10 X has a separable dense path component, 2. There exists a pseudocone P(X) and an embedding f; P(X) e x x I such that f(X) = x X {0}. 39 There exists a pseudocone P(X) such that X is a retract of P(X). Chapter III" Pseudocones over metric spaces Proposition: Every pseudocone over a metric Space is metrizablea Theorem: A Peano continuum X is a retract of every pseudocone P(X), David Parham Bellamy Chapter IV° A generalization of Peano continua Peano continua are characterized in terms of pseudocones as are compact metric continua X with the property that 2 W,° a) X = U W. where each W. is a Peano continuum and W. . 1 1 1+1 1 1=l Chapter V. The Stone-CEch compactification of A e The base of B(A) is called A30 Proposition 1: Every metric continuum and every pseudo- cone is a continuous image of every nondegenerate subcontinuum * of A 7% Corollary: Every nondegenerate subcontinuum of A has . c cardinal number 2 ‘9“: Proposition 2: A is an indecomposable continuum. TOPOLOGICAL PROPERTIES OF COMPACTIFICATIONS OF A HALF-OPEN INTERVAL By David Parham Bellamy A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1968 J 53073 I'- / ‘ ,2 , Q (3“ ACKNOWLEDGEMENTS The author wishes to express his appreciation to his major professor, H.S. Davis, for his many helpful suggestions and criticisms on this and other problems. He also wishes to thank all the other faculty members and graduate students in the Department of Mathematics with whomhe has had stimulating discussions during the investigation, ii TABLE OF CONTENTS ACKNOWLEDGEMENTS INTRODUCTION CHAPTER I Existence and General Properties CHAPTER II Retracts of Pseudocones CHAPTER III Pseudocones over Metric Spaces CHAPTER IV A Generalization of Peano Continua CHAPTER V The Stone-CEch Compactification of A APPENDIX Unsolved Problems Suggested by this Thesis REFERENCES Page 13 23 30 45 48 INTRODUCTION In this thesis, A will always denote the half-open inter- val [l,w). Let (S,h) be a Hausdorff compactification of A where h: A v S is the natural embedding and let X = S-h(A). It is the purpose of this thesis to give partial answers to the following questions: What properties must X possess? What relationships exist between the topological properties of X and those of S? This point of view motivates the following definitions: A compactification (S,h) of A is a pseudocone: S-h(A) is the base of S; h(l) is the vertex of S; and if the base of a pseudocone S is homeomorphic to a topological space X, S is a pseudocone over X. The notation P(X) will be used to denote a pseudocone over X whenever it is useful. This is not meant to imply any sort of functorial relationship. Unless explicitly stated otherwise, both A and X will be considered identified with their images in a pseudocone P(X). The first chapter is devoted to general properties of pseudocones and proofs of the existence of pseudocones over certain classes of compact Hausdorff continua: In the second chapter the question of when X is a retract of P(X) is inves- tigated. The third chapter discusses pseudocones over metric spaces; it is shown that every pseudocone over a metric space is metrizable, and machinery is developed to prove a stronger result about retractions in this case° In chapter four, a character- ization of Peano continua in terms of pseudocones is given, and a class of metric continua more general than Peano continua is investigated in some detail: The fifth and final chapter is devoted to an examination of some of the continua-theoretic properties of the Stone-CEch compactification of A, CHAPTER I EXISTENCE AND GENERAL PROPERTIES The following two lemmas give the most important pro- perties shared by all pseudocones. Lemma 1: A pseudocone P(X) isflg continuum irreducible be- tween two points a and b, where a is its vertex and b is any element of X. Proof: Let U be any Open Subset of P(X) missing a and b: A is dense in P(X), so U H A # ¢° Let y E U O A. Since A is locally compact, A is open in P(X). Therefore, [1,y) is open in P(X)° Also, [1,y] is closed in P(X). Let B = [1,y) fl (S-U)a Since y E U, B = [1,y] O (S-U) B # ¢ since 1 = a E B. B # S-U since b E B: Thus B is a nonvoid proper closed and open subset of S-U° Hence S-U is not connected and the proof is complete: Lemma g: The base 2£.§ pseudocone is.a compact Hausdorff continuum. Proof: Let P(X) be a pseudocone: For each positive integer n, let An = [n,®) and set Pn(X) = An U X. Then >< u 38 1 Pn(X) n But Pn(X) = Ag, so that each Pn(X) is a compact Hausdorff continuum. The intersection is monotone, so X is also a com- pact Hausdorff continuum. The principal results concerning the existence of pseudo- cones follow: Theorem I: Let X 23.3 compact Hausdorff continuum which is irreducible about some separable subset B pf itself and such that there exists a locally path connected locally compact Hausdorff space Y and a3 embedding f: X r Y such that f(X) is a G5 subset g: Y. Then there exists 3 pseudocone over X which embeds 12 Y X I with base f(X) X {0}. Proof: Let X be identified with its image in Y and GD let i-l be a sequence of open sets in Y such that: 1) U1 is compact ' Q 2) for each 1, Ui+1 Ui 3) O U = X 4) each Ui is path connected: Let D be a countable dense subset of B and let :=1 be a sequence from D such that for each x E D, the set NX = {izai = x} is nonvoid and Nx is infinite if x is an isolated point of Do Note that every neighborhood of any point in B contains elements from of arbitrarily large index. For each i, let wi: [i,i+l] * Ui be a path from ai to CD a, : Set w = U w,: Then w is a continuous map from A into 1+1 i-l 1 _ * — _ U1. Define w : A a U1 X I (where I = [0,1]) by: * 1 w (t) = (0302),? * Then w is an embedding of A into U1 X I: Identify Ul with 'U1 X {0}. Since U1 X I is a compact Hausdorff Space, * w (A) is a pseudocone: It remains to be shown that X is the * 2 base of w (A). Let X be the base of w7(A): Let p be any 0 element of (U1 X I) - (X U w*(A)). If p Q U1, p = (x,t) and (‘61 x (5,11) - «3(th is an open neighborhood of p in U1 X I which misses w*(A). If p 6 U1, let V be an open neighborhood of p in U1 such that V O X = ¢- Then {(Ui - Ui)}:=2 is an open covering of V and by compactness there is an n such that VIC U1 - Un“ Thus, V 0 Un = ¢: V X [O,%) is an open neighborhood of p in U1 X I. Suppose * l x 6 w (A) n (v x [0,?) < , t > n. Therefore w(t) E Un ('le I’ll-4 Then x = (w(t),%). Since e and w(t) i V, a contradiction. Thus, w7(A) m (V X [O,%)) = ¢ Q and X0 X. Now suppose p 6 Bo Let V be any open neighborhood of p in Ul X I. V contains a neighborhood of p of the form W X [0,6) where W is open in U1 and 5 > 0. By the choice . l of the sequence , there eXists an n >'— such that a E W. E n * * Then w (n) = (an,%) and w (n) E W X [0,5) E V. Therefore, B Q X0. By Lemma g, X0 is a continuum, and since X is irre- ducible about B, X0 = X. This completes the proof. Corollary 1: _£’ M i§.a compact metric continuum, there exists a pseudocone over M. Proof: M is itself separable, and embeds in the Hilbert Cube as a G set. 6 Corollary 2: If X .l§.E compact Hausdorff continuum which con- tains a separable dense path component D, there exists a pseudo- cone over X. Proof: Let {a1}:=1 be a countable dense subset of D and let w: A r X be a continuous map such that for each positive integer i, w(i) = ai. This is possible Since D is path con- nected. * Define w : A r X X I by 7': 1 w (t) = 0002),?) Then w*(A) U X X {0} is the desired pseudocone. Since a pseudocone is separable and completely regular, it embeds in IC. This implies that the base of a pseudocone has cardinality at most 2C. Corollary 2' implies that there exists C . . . . pseudocone over I , showing that this cardinality can be realized. CHAPTER II. RETRACTS OF PSEUDOCONES The question of what properties a continuum must possess to be a retract of every pseudocone over itself appears to be difficult. A simple result is given here and a more general result is established for the metric case in Chapter III. Proposition 1: ‘Ap absolute neighborhood retract X lg §_£g- tract pf every pseudocone P(X). lgpppfz Let P(X) be a pseudocone. Since X is an absolute neighborhood retract, there is an open neighborhood U of X in P(X) such that X is a retract of U. P(X) - U is a compact subset of A and hence there is a real number x such that P(X) - U C [1,x). Let r: U r X be a retraction and define g: P(X) * U by: g(t) X if t E [1,x] t if t E [1,x) Then rag: P(X) e X is the desired retraction. The question of whether, given X, there exists a pseudo- cone P(X) such that X is a retract of P(X); and whether, given P(X), X is a retract of P(X) are simpler, and char- acterizations are given for these cases. Lemma 1: Given a pseudocone P(X), X .l§.§ retract pf P(X) if .gpg,gply if there l§.§E embedding f: P(X) m X X I such that f(X) = x X {o}. 2392:: If P(X) can be embedded in X X I in the pre- scribed fashion, then f-lopof is the desired retraction, where p: X X I m X X {0} is the projection. Conversely suppose X is a retract of P(X). Let r: P(X) H X be a retraction and define P(X) r I by: f0: f (x) .l if x e A O x = 0 if x E X Let f: P(X) * X X I be defined by: f(x) = .f0> Since r and f0 are both continuous, so is f. To show f is one to one, suppose f(x) = f(y). Then fO(X) = fO(y) so that x = y if either x E A or y E A. Thus suppose x,y E X. Then r(x) = r(y) and x = y since r is the identity on X. The compactness of P(X) and the Hausdorff property of X X I make f an embedding, and the proof is complete. Corollary 1: Let P(X) pg 3 pseudocone. There exists pp em- bedding f: P(X) a x x I such that f(X) = x x {0} ii and only if there exists a topological space Y, a point y E Y, and an embedding g: P(X) m X X Y such that g(X) = X X {y}. Proof: Let r: X X Y r X X {y} be the projection. Then -1 g orog is a retraction from P(X) to X. Thus, by Lemma 1, there is an embedding f: P(X) r X X I with the desired pro- perty. The converse is clear. Lemma 2: Let X 23.3 topological space with a_dense path com- ponent D and f: X r Y 3 continuous surjection. Then Y has a dense path component C, and C lg separable if D is. Proof: It is clear that the path component of Y con- taining f(D) is dense and is separable if D is. Theorem I: Let X 22.3 compact Hausdorff space. Then the follow- ing three conditions are equivalent. 1) X has 3 separable dense path component. 2) There exists 3 pseudocone P(X) and £3 embedding f: P(X) a x X I such that f(X) = x X {0}. 3) There exists a pseudocone P(X) such that X i§.a retract pf P(X). Prpgfi: 1) implies 2) by the construction used in proving Corollary 2, Chapter I. 2) implies 3) by Lgmma l. 3) implies l) by Lemma 3 and the separability of A. Corollary 3: Let X pg 3 compact metric Space. Then the following three conditions are equivalent. 1) X has a dense path component. 2) There exists a pseudocone P(X) and 33 embedding f: P(X) e x X i such that f(X) = x X {0}. lO 3) There exists 3 pseudocone P(X) such that X is a retract pf P(X). It seems natural to ask whether the hypothesis f(X) = X X {0} is necessary to prove that X is a retract of P(X) in LEEEE.A° Can this hypothesis be eliminated or perhaps weakened to f(X) C X X {0}. The following example shows that this is not possible. Example 1. In E3, Euclidean 3-space set: x={ (2.2.2). and let f be the restriction of f0 to S. Then f is an em- bedding and f(W) C W, but by Corollary 2, W is not a retract of S since W has no dense path component. In view of the above result concerning dense path components of continua, it may be reasonable to ask whether path connectedness 11 of X is sufficient to imply that every P(X) retracts onto X. This is not the case, for it is fairly easy to construct a pseudo- cone over a Warsaw circle which cannot retract onto its base as follows: Example 2: In E3 let: B = {(x,y,0): y = sinl and O < x S l} x n c = {(0.y.0>: -2 s y s 1} l D = {(x,-2,0); 0 s x : -2Sys0} Set X = B U C U D U E. Then X is a Warsaw circle. Construct a pseudocone over X as follows: For each positive integer i, let: 1 l 1 Di be the segment from (U’-2’i) to (O,-2,i+1) l = — ° - S S l l = _ . g 3— Ni {(X’O’i+1)° O x in} R={(xy——1):(xyO)EB and xz—l} i ’ ’i+l ’ ’ in l l = — _ ‘ - S S E1 {(n’y’i+l ' 2 y 0}“ Set B, = R, U N, for each i. Then i i 1 co P(X) = X U I U (Bi U Ci U Di U Ei)] is a pseudocone over X, i=1 and, roughly speaking, each Di is over D running from the Hdha level to the z = Tl— level, each Bi’ Ci’ and Bi is z = 1+1 12 over, B, C, and E, respectively, at the z = Til level. To show that X is not a retract of P(X), let: U = {(X.y.0) E X: y > ~2} 5 V: {(033730) EX- '_‘EE and y > -Z} For each ositive inte er n let = (—l —l- and p g ’ pn n’ ’n+1 3 l = _ _.___ g let qn (O, 2’n+l)° Let IH CH U Bn be the are from qn to pH and let — (l O O) - l' P TT, 9 " 1m pn n—m a d - (O - 2'0) - l' n q — , 2, — im qn n—m Now suppose r: P(X) * X is a retraction. Then: lim r(pn) = p and lim r(qn) = q. new new Thus for some sufficiently large integer k, r(pk) E W; C r-1(U). Thus r I is a path in U from r(qk) E V; and I k k r(qk) to r(pk), which is impossible since r(qk) and r(pk) lie in different path components of U. CHAPTER III PSEUDOCONES OVER METRIC SPACES In this chapter, certain relationships between pseudocones and compactifications of closed subsets of A are studied. Many of the techniques used rely upon the metric structure, but some of the questions involved are applicable to the non-metric case. These are treated briefly in Chapter V. Definition 1: Let B be a closed noncompact subset of A and let B' be a compactification of B. Identify B with its image in B'. Let P(X) be a pseudocone. B' extends £9. P(X) if and only if there exists an embedding f: B' r P(X) such that f(B' - B) = x and le: B e A is the inclusion map. If such a pseudocone exists but is not specified, B' will be said merely to extend £9 a pseudocone. If B' extends to P(X), then P(X) naturally contains B'. The following result will be of use in some of the develop- ment . Lemma I: Let B .pg 3 closed unbounded subset pf A and let B' = B U Y where both B' and Y are compact and B O Y = ¢. Then B' 13 metrizable if and only if Y l§° Proof: If B' is metrizable, Y is since Y Q B'. If Y is metrizable, Y is separable since it is compact. Thus there exists an embedding fO of Y into the Hilbert Cube l3 14 C. Since C is an absolute retract, f extends to a map 0 f1: B' r C. Define f2: B' r I by: f2(x) = i if x E B x 0 if x E Y Let f: B' r C X I be defined by f(x) = (f1(x). f2>. Then f is an embedding and B' is metrizable since C X I is. . . . + . . Of particular interest is the case B = Z , the pOSitive integers. The principal questions investigated are: . . . . + 1) Given a compactification X of Z , when does X extend to a pseudocone? 2) Given a pseudocone S, when does S naturally contain . . . + a compactification of a copy of Z ? The first question is answered as follows: + Theorem I: Let X RENE metric compactification pf Z and let i: 2+.“ X 23 the natural embedding. A necessary and sufficient + condition that X extend Ep‘a pseudocone ii that X - i(Z ) pg connected. Proof: The necessity is clear from Lemma_2, Chapter I. . . . . + . . + To prove the SuffiCiency, identify Z With 1(Z ), let + Y = X - Z , and suppose Y is connected. Let f: X r C be an 15 embedding where C denotes the Hilbert Cube. Let d be the usual d(f(n),f(Y))- metric on C as a subset of Hilbert space, and let an Then lim an = 0. Set n—m Un = {P E C: d(P,f(Y)) < 2 max(an,an+1)l + For each n E Z , let wn: [n,n+l] r Un be a map such that wn(n) = f(n) and wn(n+l) = f(n+l). This is possible since the convexity of C and the connectedness of Y imply that each Un is path connected. Let Then w: A r C is a continuous map. Define g: A U Y r C X I g(x) .;l> if xeA (f(x),O) if x E Y Then gIX is an embedding since w and f agree on 2+. It is also clear that gIA is an embedding and that the complement of g(A) in the closure of g(A) lies in C X {0}. But if p E C X {0} and p E f(Y) X {0} there is an n such that p E Uk X {0} for every k 2 n. Let j 2 n be an integer such that aj 2 ak for every k 2 n. Such a j exists since the ai's tend to zero. Then p E Uj and — l (c - UJ.) X [0.3) 16 is a neighborhood of p missing g(A). But Y is contained in + the closure of g(Z ) and hence in that of g(A). Therefore g(A U Y) is the desired pseudocone. The central role of connectedness in the preceeding theorem makes the following result of some interest. Proposition 1: Let ‘pg a compact metric space and let a) < > C C . = ai i=1 BE E.§ESESEEE.2£.221E£§.IE M- If 1:: d (ai’ai+l) 0’ then the set F pf cluster points pf 13 connected. Proof: Suppose not. Let F = K.U L sep. Since F is closed, K and L are compact and d(K,L) > 0. Let d(K,L) = 38. Set (2‘ II {x E M: d(x,K) < e} {x E M: d(x,L) < e} .< ll Then U U V contains all but finitely many of the a1 and each of U and V contain infinitely many of the ai. Choose L such that, for each integer r 2 L, d(ar,a ) < E and ar E U U V. r+l Without loss of generality assume a E U. Then there exists a E smallest integer n greater than L such that an E V. Thus, d( an) > 5, contradicting the choice of &. Hence F is a n-l’ connected and the proof is complete. + Corollary 1: Let Z+IU Y 23.3 metric compactification pf Z + with metric d, where Z O Y ¢. Suppose d(n,n+l) tends £2 zero as n tends to infinity. Then Y is.a compact metric continuum. 17 Proposition 1 also has the following peripheral corollary: Corollary 2: Let , 1 1 be a sequence from M. Then con- : —"_’ _— 1 —- verges LE and only if lim d (ai,ai+1) = O. i—m The second question if also readily answered. Theorem 2: Let X pgna compact metric continuum and P(X) ‘3 pseudocone over X. Then P(X) naturally contains a compacti - . . + ication of'a copy pf Z . Proof: Let {ai}:_l be a countable dense subset of X. For each i E Z+ let i 3) For each i the sequence Cj if and only if i > j 3) C = C U Y, where 6' denotes the closure of C in P(Y). Parameterize A so that Cn = n for each n E Z 00 The set {aili—l is constructed as follows: Define a1 = 1. Suppose an_1 has been defined. Define kn to be the largest element of the set + - s {L E Z . L an-l} Set _ . 1_ Dn _ {y E A- d(Y>an_l) S n} = . S pn sup{t. [an_1.t] Dn} 19 Now define a = min(p ,k + l). n n n on Then i is an increasing sequence of elements of A. =l To show that is unbounded, suppose not. Let r be the Smallest integer greater than every ai. Let a = sup{ai}. Then a E (r-l,r] and a = lim a, ism 1 By the definition of convergence and the Cauchy criterion, there exists an integer nO such that for every n > n , both an > r-l and d(an,an+l) < ;%I° But an+1 = pn+l’ andO kn+1 = r-l, and if d(pn+1,an) < ;%I’ there is a closed neighborhood 8 = [pn+l ' 8’ pn+1 + a] such that S C Dn+l° Then [an’pn+l +.€] ; Dn+l’ contradictin the choice of . g pn+1 00 Therefore is an unbounded sequence. Let B = {ai}i-l° + . Since B contains Z , B contains Y. Since B is closed in A, E = B U Y, It is clear from the construction that lim dia ([a 11—“ n’an+l]) = 0, so the proof is complete. The preceeding results will now be applied to the proof of the following Theorem. Theorem 2: Let X EE.E Peano continuum, P(Y) a pseudocone, and f: Y r X ‘3 continuous map. Then X ig’a retract pf P(Y) Uf X. 20 Proof: Let P(Y) Uf X = R. Then R is metrizable by Lemma 1. Let d be a metric on R. The closure of A in R is a pseudocone, and by Lemma 2, A contains a set {3i}:-l such that: l) a = l 2) ai > aj if and only if i > j 3) {ai} is unbounded 4) lim dia ([an,an+l]) = 0. 11—00 , + Parameterize A so that an = n for each n E Z . Now let {bn}:-1 be a set of points in X such that for each n E Z+, d(n,bn) = d(n,X). This is possible since X is compact. Note that: O I/\ ’ d 11m (bn,bn+ n-’°° 1), IA 11m d(bn,n) + 11m d(n,n+l) + lim d(bn+ ,n+l) n—@ 11—“ 11—0) 1 = 0 And thus lim d(bn,b n-'°° n+1) = 0° Now X is uniformly locally arcwise connected, and thus, given 8 > 0, there exists a 5 > 0 such that if x,y E X and d(x,y) < 6, then there exists a path w: I r X from x to y 21 such that dia (w(I)) < E. (For brevity this will henceforth be written "a path of diameter < 3”.) + For each n E Z let = > . 3 RH {t 0. whenever x,y E X and d(x,y) d(bn’bn+1)’ there exists a path from x to y of diameter less than t}, and define En = 2 inf Rn. Then from uniform local arcwise con- nectedness and the fact that d(bn,bn+ ) tends to zero it follows 1 that lim 6 = 0 n n—OCD + Now, for each n E Z , let rn: [n,n+1] r X be a path of diameter less than e such that r (n) = b and r (n+1) = b n n n n n+1' 00 Let r = U r.. Then r is a continuous map from A O . i 0 1:1 to X Define r: R r X by r(x) = r0(x) if x E A = x if x E X Since r A and r X are continuous, to prove the con— 00 tinuity of r it suffices to prove that if i=1 is a sequence of points from A converging to p E X, then lim r(pi) = p also. To show this, for each positive integer i—m n let qn be the largest integer less than or equal to pm. Then 22 lim d(qi,pi) = O 1"“ and lim d(qi,bq ) = 0 1*@ 1 But pi E [qi,qi+l] and lim dia (r[q1,qi+l]) = 0. Therefore l—bm 11m d(r(pi),r(qi)) = lim d(r(pi,bq ) = 0 since r(qi) = bq,’ l—m l—m 1 1 Thus 0 S lim d(p..r(p.)) ifim i i s . . . 11m d(pi,qi) + 11m d(qi’bq,) + 11m d(bq',r(pi)) 1am 1mm 1 I‘m 1 = O or l—>oo and thus lim r(pi) = lim p, = p l—aoo i—ico 1' and the proof is complete. Corollary 3: Let X pg‘g Peano continuum and P(X) a pseudocone over X. Then X i§.a retract pf P(X). Proof: Let f be the identity map on X. Corollary 4; Let X 22.3 Peano continuum and P(X) ‘3 pseudocone over X. Then there exists 33 embedding f: P(X) r X X I Such that f(X) = x x {0}. Proof: Clear from Lemma 1, Chapter II. CHAPTER IV A GENERALIZATION OF PEANO CONTINUA In view of the above results on dense path components of bases of pseudocones, it is natural to ask what other properties of continua related to path connectedness can be examined in terms of pseudocones. Definition 1: A pseudocone P(X) is said to be even if and only if there exists a retraction r: P(X) r X and a continuous map h: A r A such that, for each t E A, h(t) > t and r(t) = roh(t). Lemma 1: Let h: A r A 23.3 continuous map such that, for each t E A, h(t) > t. Then h(A) = fps”) for some p E A. Proof: It is clear that h(A) is an unbounded interval. Let p = inf h(A) and let q > p. Then if t > q, h(t) > q, so that p = inf h[1,q]. Since [1,q] is compact, p E h[1,q] C h(A), and the proof is complete. The Hahn-Mazurkiewicz Theorem now yields a characterization of Peano continua in terms of pseudocones. Theorem 1: Let X 23.3 topological space. X l§.3 Peano continuum if and only if there exists an even pseudocone over X. Proof: Suppose X is a Peano continuum. Let f0: I r X be a continuous surjection such that fO(O) = f0(l). Such a map exists by the Hahn-Mazurkiewicz Theorem. Define f: A r X X I by 23 24 f(t) = (f (t-Itl) 1) O ’t where [t] denotes the greatest integer function of t. Set P(X) X X {0} U f(A). To see that P(X) is an even pseudocone, let r: P(X) r X be the restriction to P(X) of the projection X p: x I a x X {0} and define h: A a A by h(t) = l+t. Conversely, suppose P(X) is an even pseudocone. Let r: P(X) m X and h: A r A be the given maps. r(A) is dense in X since A is dense in P(X). Let h(A) = [p,m). Suppose x E r(A) and let y = inf (rIA)-l(x). Since (rIA)-1(x) is closed in A, y E (rIA)-1(x). It follows that y E [1,p], for if not, there exists t E A such that h(t) = y. Then r(t) = roh(t) = r(y). This is a contradiction to the choice of y, since t < y. Thus, x E r([1.pl). and r([l,p]) = r(A). Therefore r([l,p]) is dense in X. But r([l,p]) is compact and hence closed. Thus r([l,p]) = X, and X is a Peano continuum by the Hahn-Mazurkiewicz Theorem. The preceeding proof rests heavily upon the fact that a Hausdorff space is a Peano continuum if and only if it is a con- tinuous image of a closed interval. A generalization of this notion is the following: 25 Definition 2: A compact metric space X is an almost-Peano con- tinuum if there is a continuous surjection f: A m X. This is a proper generalization of Peano continua since the Warsaw circle is a continuous image of A. A concept related to this is that of almost local con- 2 nectedness. A topological Space is almost locally connected if and only if every open set contains a connected open set. This property is investigated below in some detail and leads to a characterization of almost-Peano continua. Lemma 3: A pseudocone is almost locally connected. Proof: Every open set in a pseudocone meets A and hence contains an open interval. If X is an almost-Peano continuum and f: X r Y is a continuous surjection where Y is a compact Hausdorff space, then clearly Y is an almost-Peano continuum. However, the property of being almost locally connected is not preserved by continuous maps in general. Example 1: Let X be the cone over the Cantor set. By Corollary 2, Chapter II, there exists a pseudocone S over X such that X is a retract of S. Let r: S r X be a retraction. Then r is a continuous surjection, and S is almost locally connected by Lgmma.2. However, X is not almost locally connected since X with its vertex deleted contains no connected open set. 26 The following two results establish some conditions under which almost local connectedness is preserved. Proposition 1: Let X and Y _pe topological spaces and let f:.X r Y ,p; a continuous ppen surjection. __f X .ip almost locally connected, pp lg Y. Proof: Let U be an open subset of Y. f (U) is open in X and hence contains a connected open set V. Then f(V) is an open connected subset of U in Y. Proposition 2: Let S pglg locally compact Hausdorff space and suppose S .l§.é countable union_gf closed, almost locally con- nected subsets. Then S lg almost locally connected. Iggppf: Suppose S = .81 Wi where Wi are closed and al- most locally connected. Letl_U C S be open. Since S is locally compact, U contains an open set V with compact closure. Then l(V n wi) .-t->. XX {0} and A with f(A) and set P(X) be the projection and define r: P(X) r X rIA = f so that rIA is a surjection as 0 implies 1) This is clear since r A: A a X is the required map. = X U A. re- CHAPTER V THE STONE-CECH COMPACTIFICATION OF A Let X be a completely regular topological space. The following notation will be used: B(X) is the Stone-Céch com- pactification of X and X is considered to be a subset of B(X). If Y is a compact Hausdorff space and f: X r Y is a continuous map, fl is the unique extension of f to B(X). B(X) - X is denoted X*. The following lemmas will be useful in some of the develop- ment. Lemma 1: Let P(X) pg_g pseudocone and let U pg gp open set which meets X. Then U O A _lg unbounded. Proof: Suppose not. Then U O A C [1,X] for some x E A. Thus, is a nonempty open subset of P(X) which misses A. This is impossible Since A is dense in P(X). Lemma 2: Let P(X) pg.g pseudocone and let i: A m P(X) pg the inclusion map. ll f: A r A 13.3 continuous map such that 7‘: 7"“ f(t) see, as t -+ CD, then iof (A) = x. * Proof: (iof) (B(A)) is a closed subset of P(X) con- taining (iof)(A). Since 30 31 (iof>‘c 3% (iof) (B(A)) = (i°f)(A) U (i°f) (A ) A A = iip.°°> U (iof) (A >. Thus x : (iof)*(A*). A ' e To prove (iof) (Ax) C X, suppose x E A7 and 7': (iOf) (x) E i(A). Let U be an open set in B(A) such that: — 7‘s-l x e U C U ; (iof) (i(A)) _ e _ Then i 1((iOf)7(U)) is a compact subset of A and hence bounded. Therefore there exists a y E A such that for every t > y, .1 7‘: f(t) > SUP i ((i°f) (U))~ But for t E A, f(t) = i'l<(iof>(t>> -1 A i ((i°f) (t)) 32 and thus for t > y, -1 '2': -1 7% i ((i°f) (t)) > SUP i ((iof) (U)) A Therefore U O (y,m) = ¢, and by Lemma l, U O A = ¢. But A x E U O A , a contradiction, and the proof is complete. The importance of the Stone-CEch compactification of A to the study of pseudocones stems from the following result. Proposition l: Let X .pg_g Hausdorff space. Then there exists 0.. g pseudocone over X if an only ll X is 3 continuous image of A Proof: Suppose P(X) is a pseudocone over X. Let , 7': 7': A i: A m P(X) be the inclu31on map. Then (ioLA) IA maps A onto X by Lemma 3. 7's“ Conversely, if f: A r X is a continuous surjection, then B(A) Uf X is the desired pseudocone. An examination of continuous images of other spaces asso- ciated with B(A) may also be of interest. In what follows, a simple characterization of continuous images of B(A) is given, and some information is obtained concerning continuous images of 7': nondegenerate subcontinua of A . A complete characterization in the latter case appears to be difficult. 33 Proposition.2: Let X .22 g compact Hausdorff Space. Then t e following three conditions are equivglent: l) X islg continuous image of B(A). 2)“ There exists g pseudocone S ._f which >< (D Is 0 o n u tinuous image. 3) X has g separable dense path component. ‘nggfiz 1) implies 2) This is clear since B(A) is a pseudocone. 2) implies 3) S has a separable dense path com- ponent. Thus by lgmmg‘l, Chapter II, so does X. 3) implies 1) Let D be a separable dense path component of X and let {ai}:=l be a countable dense subset of D. Since D is path connected, there exists a map f: A m D such that for each positive integer n, f(n) = an. Then f(A) is dense in D and hence in X. Thus f*(B(A)) is dense in X, and Since f*(B(A)) is compact, f*(B(A)) = X, and the proof is complete. Lemma 3: Suppose X lg g separable compact Hausdorff continuum. Then there exists g subcontinuum M ‘gl X X I such that M .lg irreducible between two points and X X {0} C M. co Proof: Let i be a sequence of points in X such =1 on that the range of is dense in X. Let {Lili-l be a collection of subcontinua of X such that for each i, Li is co irreducible between a, and a, . In X X I define {W,}, 1 1+1 1 i=1 34 co and {b,}, as follows: 1 i=1 W2k-l = Lk X If} W2k ={arm} [Ti—1'51 b2k-l = (akfi) b2k = (ak+l"i) for each positive integer R. Then for each i, Wi is irreducible between bi and bi+ and Wi O Wi+l = {bi+l}° Set 1) oo M= U w.U (XX{0}) i=1 1 co Then M is the closure in X X I of U Wi and hence a continuum. i=1 To prove that M is irreducible between two points, let x E X X {0} and suppose F is a proper closed subset of M con- taining both b1 and x. It remains to be shown that F is not connected. There are two cases to be considered: 1) If for some i > 1, b1 E F, then i-l U (W. O F) i=1 3 is a nonvoid proper closed and open subset of F. co 2) If each bi E F, then since U W1 is dense in M, i=1 there exists some Wi which meets M - F. Then F O Wi can be expressed as the disjoint union of two closed sets, B and C, such that b. E B and b, E C. Then 1 1+1 35 i-l U (w. n F) U B i=1 J is a proper nonvoid closed and open subset of F. Thus F is not connected and the proof is complete. Corollary_l: Suppose X is_g separable compact Hausdorff con- tinuum. There exists g compact Hausdorff continuum M containing X such that X is g retract pl M and M _lg irreducible pp- tween two points. Proof: Identify x with x X {0} in x X I, and let M be as in Lemma 3. A retraction is obtained by restriction to M of the projection p: X X I r X. Corollary 2: Let X pg‘g compact metric continuum. Then there exists g compgct metric continuum M containing X Such that X lg_g retract pl M and M .lg irreducible between two points. Proof: Let M be as in Corollary l. M is metric since X X I is. Lemma 4; Let U and V pp unbounded open subsets pl A such that U O VI= ¢ and inf U < inf V. Then there exist seguences Q co (1) co

, , and such that: n n=l n n=l n n=l -——— n n=l or each ositi e inte e n < < r < s < . 1) F p V g r ’ pn qn n n pn+1 w m g Q 2) U U [pn,qn] and V U [rn,sn]. n=l n=l Indication pl_proof: Define 36 p1 = inf U q1 = sup {t E U: [p13t] O V = ¢l r1 = inf V 31 = sup {1: E V: [r1,t] n U = d} Then pn, qn, rm, and SH are defined inductively, replacing U and V with U O [Sn-1,“) and V O [s respectively, n-l’m) in the definition of pH and rn and replacing p1 and r1 by pH and rn respectively in the definition of qn and Sn Verification of the properties 1) and 2) is then an elementary exercise in real analysis. Lemma 5: Suppose M is a metric continuum irreducible between A two points, a and b, and suppose W is a subcontinuum of A and x and y are distinct elements of W. Then there exists a continuous surjection g: W e M such that g(x) = a and S0?) = b. Proof: By Corollary 1, Chapter I, there exists a pseudo- cone P(M). Let i: A m P(M) denote the inclusion map. By Lemma 1, Chapter III, P(M) is metrizable. Thus there exist and :- of elements of A such that: 00 se uences q n —1 n=1 ' ' ' < < . 1) For each pOSitive integer n, an bn an+1 2) lim i(an) = a and lim i(bn) = b. 11"” I‘l"’°° Now let U and V be open sets in B(A) Such that 1) x E U and y E V 37 2) finV=¢. 3) inf (U O A) < inf (V O A). By Lemma 1, U O A and V O A are unbounded, and by Lemma . m m m m 4 there eXists sequences and n=l’ n n=l’ n n=l’ n=l SUCh that: 1 For each ositive inte er n < < r < S < p . on Q 2) unA: U[p,,q,] and VOAC UEr,,s,]. i=1 1 1 i=1 1 1 Define f: A v A as follows: Set f(1) = l f(t) = an for t E [pn’qn] f(t) = bn for t E [rn,sn] and extend f linearly to each of the intervals [1,p1], [qn,rn], and [s Then since the sequences ’ ’ , and n’pn+l]° are unbounded, f(t) is defined for each C E A and f(t) H m k A A as t a w. Therefore, by Lemma 2, (iof) (A ) = M, and (iof) (W) is a subcontinuum of M. A-1 m Now (iof) ({i(aj)}j_1 U {a}) is a closed set containing U O A, and hence containing the closure in B(A) of U O A. Let K be any open set in B(A) containing x. Then K O U is an open set containing x, and since A is dense in B(A), K O A O U = ¢° Therefore, x belongs to the closure of U O A in B(A), and hence (iof>* 6 {i}?=1 U {a} 38 But (iof)*(x) E M and {i(aj)}:=l O M = ¢° Thus (10f)*(x) = a. By a parallel argument, (iof)*(y) = b. Then (iof)*(W) is a sub- continuum of M containing both a and b, and Since M is irreducible between a and b, (iOf)*(W) = M. Setting A g = (iOf) IW completes the proof. Lemma p: Suppose 8 .lg g pseudocone and W lg.g nondegenerate 7': subcontinuum of A . Then there exists g continuous surjection h: W m S. Proof: Let x and y be distinct elements of W and let U and V be open sets in B(A) such that: l) x E U and y E V. 2) ‘U n V = e. 3) inf (U n A) < inf v n A. co co co co > > > <> Then let > n v n A c [1,sn]. Therefore g(y) belongs to the base of S, and by Lemma l, Chapter I, S is irreducible between g(x) and g(y). Since g(W) is a subcontinuum of S containing both g(x) and g(y), g(W) = S and g W is the desired map. Theorem l: Let X pg g.compact Hausdorff continuum and let W A pp p nondegenerate subcontinuum pl A . f~ X lg metrizable pl ll, X has g separable dense path component, then X lgng con— tinuous image of W. Proof: If X is metrizable, by Corollary 2, there exists a metric continuum M irreducible between two points and a con- tinuous surjection h: M m X. By Lemma 2 there exists a continuous surjection g: W H M. Thus hog is a continuous map of W onto X. 40 If X has a separable dense path component, berroposition 3 there exists a pseudocone S and a continuous surjection h: S m X. By Lemma p, there exists a continuous surjection g: W m S. Then hog is the desired map. A Corollary g: _l_ W lg.g nondegenerate subcontinuum pl A , the cardinal number pl W lg 2C. Proof: Since B(A) embeds in IC, the cardinal number of W is at most 2C. Since IC is separable and path connected, by Theorem l there exists a continuous surjection f: W m IC. Thus the cardinal number of W is at least 2C 7‘ Corollary 3: Let X pg.g compact metric continuum and f: X r A g continuous map. Then f lg‘g constant map. A Proof: f(X) is a subcontinuum of A and has cardinal number at most c. By Corollary 2, such a continuum is a single point. The remainder of this chapter is devoted to two miscellaneous topics related to B(A). The notion of natural containment was introduced in Chapter III and discussed there for metric pseudocones. In the following two results, B(A) is contrasted with certain pseudocones over continua with separable dense path components with reSpect to this property. Proposition 2: Let B pg_g closed subset pl A. Then B(A) naturally contains E .ll and only ll A - B lg bounded. 41 Proof: Suppose A - B is bounded. Then let t E A Such that: [t,oo) C B. 7': _ Now A lies in the closure of [t,m) and hence in B. But since _ A — A B is closed in A, B - B C A . Hence B - B = A no Conversely, suppose A - B is unbounded. Let n=1 be a sequence of points in A - B such that for each positive CO integer n, an > n. Then {an}n-l is closed in A. Hence by normality there exists a function f: A r I Such that f(x) 1 for x E B a: O for x E {an}n=l 7'c - Then f 1([0,1)) is an open set in B(A) missing B but meet- A A _ ing A . Therefore A is not contained in B, and the proof is complete. Proposition i: Let X pgig compact Hausdorff gpace with g separable dense path component. Then there exists g pseudocone + P(X) which naturally contains g compactification pl Z . Proof: Let D be a dense path component of X with a co countable dense subset {ai}i-l“ Let f: A r X be a continuous function such that f(n) = an for each positive integer n. 42 Define g: A a X X I by 1 g(t) = (f(t) .-—t-> and set P(X) = g(A) U (x X {0}). Now let (x,0) E X X {0}, and let U be any open set in X X I containing (x,0). Then U contains an open set of the form V X [0,5) for some 5 > O and some open set V containing x. Then m is an open set and hence meets {a l 1. n n= E and hence H'IH + and hence to U. Thus X is contained in the closure of g(Z ) and the proof is complete. There appear to be no known examples of non-metric inde— composable continua. The following result provides such an example. 7'€ Theorem 2: A lg an indecomposable continuum. A Proof: Suppose A = X U Y where X and Y are proper A closed subsets of A . It will be shown that X is not connected. Let x E X - Y and y E Y — X and let U and V be open sets in B(A) such that: l) x E U and y E V 43 co m Choose sequences co - a . a and (r ,>, from A as i=1 1 i=1 i 1 =1 follows: Let pi E U O A. Then choose ql > pl Such that q1 E V. This is possible since V O A is unbounded. Then choose r > q1 such that (q1,r1)C V. This is possible since V is open and hence ql lies in some open interval in V. Proceeding inductively, suppose pk, qk’ and rk have been chosen for k.< n such that for each k: 1) pk E U. 2) The interval (qk’rk) is contained in V. 3) and if k.< n-l, then r < p < < qk r k pk k’ k+l‘ Then since U O A is unbounded, it is possible to choose pn > rn_1 such that pn E U. Since V O A is unbounded, there exists a > h th t E V. S' V ' , a be h se qn pn sue a qn ince is open rn m y c o n ; greater than qn such that (qn’rn) V. Then the sequences , ’ and are unbounded, for if not they have a common supremum t, and it follows t E U'O V, a contradiction. Define f: A a I as follows: First set f(t) = 0 if t 3 p1 f(pi) = 0 if i is odd = 1 if i is even f(qi) = 1/3 if i is odd = 2/3 if i is even f(ri) = 1/3 if i is even 2/3 if i is odd 44 Then extend if linearly to each of the intervals [p1,qi], ]. [qi’riJ’ and [ri’pi+l *-1 Then f (O) is a closed subset of B(A) containing { }CO H f*-1 * l' ' ' f p2k+1 k=l° ence (O) meets A . But any imit pOint o { }°° 1' . — d . *‘1 p2k+1 k=1 ies in U an hence not in Y. Thus f (O) A A meets X, or O E f (X). Similarly, l E f (X). e- But let a E f7 1(l/3,2/3). Then a is a limit point of -1 f (1/3,2/3), and -1 a f (l/3,2/3) = (q ,r ) C V k k k=1 _ e Thus a e v and hence a e x. Therefore, f°(X) n (l/3,2/3) = o, and hence X is not connected. This completes the proof. t Observe, however, that X7 is a decomposable continuum if X is a half Euclidean space of dimension greater than one. Let X = {(x1,.oe,xn) E En: x 2 0} 1 >4 l l - {(x1,...,xn) E X : xn 2 O} X = {(xl,...,xn) E X : xn S 0} For each positive integer k, let R be that set of points in k A w _ X with norm greater than or equal to R. Then X = O Rk’ k=1 7': where the closure is taken in B(X). Thus X is a continuum. Similar arguments Show that X1 - X1 and X2 - X2 are continua, and the reader can easily Show that these are proper subcontinua r A _. _ of X7. But X = (X1 - X1) U (X2 - X2), completing the argument. APPENDIX UNSOLVED PROBLEMS SUGGESTED BY THIS THESIS This investigation raises several questions which are as yet unanswered. Following is a brief discussion of some of these which the author considers most important to the further develop- ment of the theory. I. Existence of pseudocones Except in the metric case and the case of spaces with separable dense path components the results on existence of pseudocones are not particularly useful. A characterization of those continua over which there exist pseudocones in terms of intrinsic topological properties without the hypothesis of the existence of an embedding space would be most enlightening. Lacking this, more readily verified sufficient conditions would be useful. For example, does separability of X imply the existence of a pseudocone over X? Another type of problem in this connection is the follow- ing: If a collection {XaiaEJ of continua is Specified such that for each a there exists a pseudocone P(Xa)’ under what con- ditions is there a pseudocone over H (X )? A similar question aEJ can be asked about inverse limits of continua. II. Retractions Even for the metric case, the problem of when a continuum 45 46 X is a retract of every pseudocone over itself is unsolved except for Peano continua. It is also not clear whether this property is preserved by continuous surjections or even by continuous monotone surjections. III. Metric pseudocones The following problem originally motivated this study: Given a totally bounded metric p on A, let S denote the completion by Cauchy sequences of the metric space . What is the relationship between the properties of p as a real- valued function and the topological properties of the base of S? A satisfactory answer to this question Still seems difficult. In view of the concept of natural containment, the same problem may be of interest for metrics on positive integers. IV. Extension of maps on pseudocones The map h: A H A, in the definition of an even pseudocone, has an extension to the entire pseudocone, and its restriction to the base must be the identity function. In a more general setting it may be asked under what conditions a map from. A to A extends to a map from an entire pseudocone P(X) to itself, and under what conditions such a map induces a self homeomorphism of X. This homeomorphism question may be of special interest in case the pseudocone under consideration is B(A). V. The Stone-Céch compactification of A One question of interest here is whether B(A) is the only 47 pseudocone over A* up to homeomorphism, and if so whether B(A) and the one-point compactification of A are the only pseudocones which are topologically determined by their bases. The proper subcontinua of A* seem to share many of the properties of A*. Is A* hereditarily indecomposable, or possibly even homeomorphic to each nondegenerate subcontinuum of itself? If not, do there exist pseudocones over all, or any, of the nondegenerate proper subcontinua of A*? Other Questions about A*, such as whether it has the fixed point property and whether its CSch homology groups are trivial may also be of interest. 3‘: Aside from the study of pseudocones, A also raises the question of whether a non-metric indecomposable continuum must have uncountably many composants, or even more than one composant, since the proof for the metric case depends upon the second axiom of countability. Also, if X is a locally compact metric space 7" - u n such that X is connected, what are necessary and suffic1ent 7': conditions that X be an indecomposable continuum? REFERENCES An interesting related result is found in: Magill, K.D., Jr. "A note on compactifications” Mathematische Zeitschrift 94 (1964), pp. 322-325. This definition is from: Davis, H.S. and Doyle, P.H. "Invertible Continua" To appear. 48 M'IIIIIIIIIIIIIILIILIII{IIIIJIIIIIIIIIflIIII“