INVERSE LIMITS OF FINITE SPACES" . Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY THOMAS EDWARD ELSNER 1 9 72 LI BRAR Y Michigan State g University 10, — .m-,.av‘.' a.” This is to certify that the thesis entitled Inveue Lmrrs 0F RNUESP'NES presented by TmMAs Emma ‘2me has been accepted towards fulfillment I of the requirements for degree in WAT-[CS admurpwofibssor Date June 29; 1912 0-7639 fugil ' L V HUAG ;" _ ‘ BUUK BINDERY INC. LIBRARY amosns ABSTRACT INVERSE LIMITS OF FINITE SPACES BY Thomas E. Elsner We make the following definitions. Let S be a topo- logical space. Subset X c S is co-dense in S iff for each s 6 S, {s} n X # ¢. Let X” = lim {Xa' f ] be the GB inverse limit space of the inverse system {Xa' faB] over I, a directed indexing set. Let Trazxco a Xa be the usual projection for each a E I. Subset X ; Xco is strongly dense in Xco iff for each a E I and xa e Xa' w;1(xa) n X ¢ ¢- For TO - space X an inverse limit space Xm is a finite resolution of X iff each Xa is a finite TO - space and X can be imbedded as a strongly dense subspace in X“. We will usually write X c Xco identifying X with its image via this imbedding. It is the purpose of the thesis to in- vestigate the properties of these TO - compactifications of X. In the following outline of principal results, X is a To - space unless otherwise specified. The numbering of the results does not correspond to the numbering in the thesis. Theorem A; Let B be a basis for X. Then there exists a finite resolution of X, written Xm(B), where I is the set of nonempty finite subsets of B. We say X is con- {D structible from basis B. Thomas E. Elsner Theorem ;: Each finite resolution of X is homeomor- phic to a finite resolution constructible from some basis B. Theorem ;: Let {Xi} be any collection of To-spaces with finite resolutions {Xi}. Then the product fix: is a finite resolution of the product IIXi . Theorem g; Let S be a compact TO - space and let P c S be a subspace such that [x 6 S: {—7 = [x]} c F. Then F is compact. Theorem g; Let XS be a finite resolution of a Haus- dorff space X. Then the following are equivalent. (1) X is compact. (2) X is co-dense in X”. (3) X = {x e X“: {*1 = lel. Theorem Qip Let XOD be a finite resolution of a compact Hausdorff space X. Then (1) X has the fixed point property iff Xca has the fixed point property. (2) X00 and X have the same Each homology groups. (3) X is a strong deformation retract of X”. INVERSE LIMITS OF FINITE SPACES BY Thomas Edward Elsner A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1972 ACKNOWLEDGEMENTS The author wishes to express his gratitude and appreci- ation to his major professor, Harvey S. Davis, for his many helpful suggestions and patient attitude during the course of the investigations which have led to the completion of this thesis. He also wishes to thank all the other faculty members in the Department of Mathematics who have gracious- ly contributed comments and questions regarding this work in various sessions of discussion. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . INTRODUCTION . . . . . . . . . . . . . . . . . CHAPTER ONE Preliminaries . . . . . . . . . . . . . . CHAPTER TWO Constructions . . . . . . . . . . . CHAPTER THREE Finite Resolutions - Existence & Properties CHAPTER FOUR Compact Hausdorff Spaces. . . . . . CHAPTER FIVE Examples. . . . . . . . . . . . . . . . . . CHAPTER SIX Algebraic Structures. APPENDIX Unresolved Conjectures. . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . iii Page 10 14 26 34 45 S3 56 INTRODUCT ION Let Xco = lim {Xa' f } be the inverse limit space of a6 finite TO - spaces, Xa' For each a. let wa:Xon 4 Xa be the restriction to XOD of the natural projection from the product space, H Xa' to the factor space, Xa' Say that subspace X : X” is strongly dense in X00 iff 7r;1(xa) nXfiQI for each a and xa e Xa' Further, XOD is a finite reso- lution of a To - space, X, iff X may be imbedded as a strongly dense subspace of X“. It is the purpose of this thesis to investigate the properties of these TO - compact- ifications of the space, X. We make the one notational convenience that the closure of a singleton (i.e. T;Tj will usually be written without brackets. The first chapter is a listing of general results and properties of inverse limit systems and spaces which are references throughout the thesis. In the second chapter an inverse system of finite TO - spaces is constructed from a general topological space, X, and any basis for X. This system is used in the third chapter to show the existence of finite resolutions. After establishing that this construc- tion is naturally present for every finite resolution, some general properties of these spaces are discussed. In chapter four the much more rigid positioning of a compact Hausdorff space in its finite resolutions is investigated. Chapter five is a collection of illustrative examples for the l previous chapters. The sixth chapter deals with some of the algebraic topological properties inherited from compact Hausdorff spaces by their finite resolutions. CHAPTER ONE PRELIMINARIES The following definitions and theorems, with the excep- tion of L and M, are generally known and are included for reference for the remainder of the thesis. The notation here differs slightly from that of [3] or [1] and [2], but the content is the same. Most proofs are not given and al- phabetic labelling is used for distinction as reference theorems. Definition: An indexing set, I, is said to be directed by a reflexive, transitive quasi-ordering, g, on I if for any pair a,B E I there is some y E I with Q,B g y. In the following, the letters I, J, K will refer to directed indexing sets which are directed by s unless other- wise indicated. Definition: Let {Xa}a€I be a collection of topologi- :X X be a con- a6 6 4 a tinuous map. Require fan to be the identity map for each cal spaces. For each pair aSB let f g _’ ' = 0 f . Th a E I and for a B s y require fay faB BY e collection of spaces and maps {Xa' faP]I is called an inverse system of spaces over I, Definition: Let be an inverse system of X . { a faB]I spaces over I. In the product space, “)%y the subspace XOD = [x 6 H Xa': a s 6 implies ) = xa] f x OLB(B is called the inverse limit space of the inverse system {Xa' f I. (By xa 6 Xa we mean the a-th coordinate of (16 x E H Xa for each a e I. That is, for Na: 11X(I 4 Xa the coordinate projection onto the factor space, we have N (x) = x .) We also use the notation X = lim {X , f I. a O. m ‘- (1 dB' The Xa are called factor_spaces of the limit space. The maps de are called the bonding_maps of the inverse sys— tem. This structure and notation may be referred to in these preliminaries and in later chapters without complete prefac- ing. Definition: Let {Xa' f 3 and [Y , g } be inverse a6 Y Y5 systems of spaces over I and J respectively. Let Y:.I» I be any map. For each y 6 J, let w : X 4 Y be a con— Y YIY) Y tinuous map. The collection of maps {mY}Y€J is an inverse system gf_maps iff whenever y s 5 in J, the following diagram commutes. f X Y(Y)Y(6) 3? X YIG) "’ .Y(y) I £06 “95 9Y6 Y(5 ._m_._m_u--m_-- -.mmiag> 'YY Usually I = J with W the identity map. NOW let X0° and Y be the inverse limit spaces of the above systems. Define i: X 4 Y as follows. For each x e X”, let a: Q ¢(x) (mY(xY(Y)))Y€J That is, if WY: nlfiy4 YY is the coordinate projection, we indicate §(x) by giving all of its coordinates. i is called the induced or limit m§p_g£ the inverse system 9§_mgp§, {my} and we write Q = lim {mY]. Henceforth Va, for each a e I, will represent the' restriction of the coordinate projection to the subspace, X , unless otherwise specified. (D Theorem.A; (i) The composition of limit maps is the limit map of the corresponding inverse system of compositions. (ii) The inverse of a limit map is the limit map of the corresponding inverse system of inverse maps. Theorem B; Let i: X 4 Y” be a limit map as above. Then Q s is continuous and unique with respect to WY c Q = mYo WY(Y). Definition: For directed set, I, J c I is cofinal in I iff for each a E I there is B e J such that a fi B. Theorem Q: Let i be given as above and let Y(J) be co- final in I. If each m , y 6 J, is an injection (bijection, Y homeomorphism) then i is an injection (bijection, homeomor— phism). Theorem D: For J c I, the system {Xa' obtained by faB}J restricting indexes to J is also an inverse system. Indi- cating the limit spaces as x: and X3, the collection of identity maps iazxa 4 Xa for each a E J induces a limit map Q: X: 4 x: called the natural projection into XJ. é sends (xa)a€I to (Xa)a€J' If J is cofinal in I then the inverse limit spaces are homeomorphic by Theorem C. Theorem E: The coordinate projections Va: X0° 4 Xa are con- tinuous and the following diagram commutes when a s B in I. /\., %-___i__.afl_____;§xa Theorem E; For each a e I let Ba be a basis for Xa' O -1 O 0 : , b Then the collection {Fa (Ud) Ua 6 Ba a E I} is a aSlS for X . m We note that by Theorem D any cofinal J g I may re- place I in this construction. Theorem G: If each X is Hausdorff then X ==lim{X , f I ._._.___.___.._. a m ‘- CL GB is a closed subspace of H)&1. Theorem H: Let each Xa be compact and Hausdorff (or finite) and nonvoid. Then Xco is compact and nonvoid (nonvoid). Theorem I: Let A c X and let A = w (A). Then {A I ‘*“"““ m a a a and (Ag? are inverse systems of subsets via the restricted bonding maps and we have —- -1 —1.—— . - A=nwa WI=0WQ (A)=1m{A‘I Theorem J: Examples (1) Let each Xa be a distinct copy of space, X, and let map fa be the identity whenever a s B. Then Xco is home- 5 omorphic to X. (2) If indexing set, I, has a maximum element, 0, then Xm is homeomorphic to XQ. The remaining results present the well known equiva— lent definition of an inverse limit space as a universal map- ping space. Initially the idea of a Hausdorff inverse limit was hoped to be useful. Theorems L and M show that nothing is gained by its definition. See Lemma 5, page 29 and Ex- ample B, page 38. Definition: Let {Xa, de] be a collection of spaces and continuous maps. Space Y with continuous maps {ga} is a left approximation of {Xa' I iff the diagram below faB commutes for a < 6. Definition: A left approximation (X, fa] of {Xa' faB] is universal over a given category of left approximations {Yi' gai] iff for each Yi there is a unique continuous map hi: Yi 4 X such that the following diagram commutes. faB \ X X XB ‘7 a ,/ at ,r' i ‘f 95 g /;P I ”- a k’ :f . . X .____-,_-_ I . of; _. . ))X Y gfix I //’fl (1 . 9.. } Ihi / 9.: Left approximation ‘~\Y '1 1 Universal approximation Theorem 5: [L, Va] is the inverse limit space of inverse system [Xa' f ] iff L is universal over the category (16 of all left approximations. Definition: Let X be Hausdorff. Left approximation (X, fa] is a Hausdorff inverse limit of inverse system {Xa' GB all Hausdorff left approximations. I iff (X, fa} is universal over the category of Theorem L: Let X = lim [X , f ]. Let subspace X c X —----——-— co :— a oo (15 be Hausdorff. Then X is a Hausdorff inverse limit of (xa, faB} iff x = xco . Proof: For each x E X”, [x] is Hausdorff. Hence there is a unique hx: {x} 4 X such that the diagram below commutes for a s B. This implies that all coordinates of x and hx(x) are the same. Hence x e X. The converse is trivial. ,X /, A 'TI'B /// Tl'a ,/’f XI: ‘15 e; Xa h 27 W;\\\\\\ X ’//%a // Theorem M: Suppose (X, fa] is a Hausdorff inverse limit of {X , f I and {X , w ] is the inverse limit. Then X 0'. (16 m C1 and Xm are homeomorphic. Proof: By Theorem L we need only show that X is im- bedded in Xm. The universal mapping property of X0° implies that there is a unique continuous map h: X 4 X“ such that the diagram below commutes. Map h is one-to-one since points are distinguished uniquely by their coordinates. If distinct points x,y e X or their images have the same co— ordinates then maps from the singletons into X as in the diagram of Theorem L are not unique. Now X0° and X are subspaces of the product H X0 and the usual projections p are such that p = f and p = n . Then the map a aI a a| a 'X X -1 °° h. : h(X) 4 X is continuous into the product since for each a , p 0 h-1 = n is continuous. a 0 X0: // "\ w / w P M x14 fflfi } x BR ,7, O. f‘ h A?» B \ / a \ ,r’ CHAPTER TWO CONSTRUCTIONS In this chapter we construct a particular inverse limit space whose factor spaces and bonding maps are defined in terms of finite collections of basic open sets from an arbi— trary topological space and the partitions that these col- lections generate. Definition: A collection u of subsets of a set X is a partition of X iff the subsets of X in u are non- void and pairwise disjoint and uu = X. Partition u pggpr erly partitions a subset F c X iff F is a union of sub- sets that are elements of u. Partition u properly refines partition u iff each subset in u is properly partitioned by y. In this chapter, X is a topological space and B is a basis for X. Let I = {a c B: a # ¢ and finite]. For each a E I define the relation (a) on X by x(a)y iff IUEOL=XEUI=IUEOL=YEUI Lemma I: (i) For each a E I, (a) is an equivalence relation. (ii) Let Xa be the partition of X induced by (a) for each a e I. Direct I by inclusion writing a g B if a is contained in B. For a s B the partition XB properly refines Xa lO 11 (iii) We will consider Xa as a point set and will write xa c X when considering xa e Xo as a subset of X. Let x e X . For each x E x c X we have a a a xa=nIUea:er}-u(Uea:xzu} Hence partition Xo properly partitions each U E a. Egggfg (i) Clear since relation (a) is based on the equality of subsets of a 6 I. (ii) This follows directly from the definition of a S B. The proof of part (iii) also helps clarify this point. (iii) Now yexa iff x(o)y iff {UEa:er]=[U€a:y€U] iff y e n [U 6 o: x 6 U} - u{U e a: x E U}. Of course we have xa = {y e X: x(a)y]. Finally for UO 6 a, x 6 U0 and xa the equivalence class of x, we have shown that xa c U by the above and so Xo properly partitions U0. For each a e I we define fa: X 4 Xa as follows. For each x 6 X, f (x) = x E X iff x 6 x g X. This natural a a a a projection is well defined and surjective by the definition of Xa as a partition. Now topologize Xo so that fa is an identification map. The resulting space is sometimes called a decomposition space for X. (See [5], p. 244) Whenever a s 6 define f ° X 4 Xa as follows. For (16' 6 each x X , f x = x iff x c x c X. This ma is BEES ae‘a’ a a a p well defined and surjective since partition X properly re- B fines partition Xa' Given all of the previous constructions we prove the following. 12 Theorem I; (i) U c X is open in X iff u x c X ---—- o a X EU is open in X. a a (ii) Each Xa is a finite TO - space. (iii) For a s B, foB is an identification map. (iv) The collection {X , f I is an inverse o o6 I system of spaces. Proof: (i) This is simply restating that the map fa is an identification map. (ii) Xa is finite since a E I is a finite subset of B. Let x , y 6 X be distinct elements. Let x E x c X a a a a and . ' ' ' = d f = . Y 6 ya ; X By definition fa(x) xa an o(y) Yo Further, since x and y are in distinct equivalence clas— ses, the definition of (o) implies that there is some U 6 a such that either x e U and y f U or x f U and y 6 U. In either case by (i) above and (iii) of Lemma 1, fa(U) is open in X and provides TO - separation for Xa a d . n ya (iii) By our definitions we have f o f = f and so o6 B o f;é = fBo f;l. Further the proper refinement of Xo given by XB implies that for each open Uo c Xa' -l -l —l f f f U = f U . B ( B( a ( o))) a ( a) Now since fa and fB are identifications, Uo is open in . -l -1 -l -l -l . Xo iff fa (Ua) — ffi (fB (fa (Ua) )) — f5 (foB (Ua)) is open . . -1 . . . . in X iff foB(Uo) is open in X6. And so fofl is an iden tification. ' F s , f = f 0 f from the defini- (iv) or o s B y GY GB BY tions. Similarly, foo is the identity on Xa for each a E I. Let Xm(B) = limea, f ] be the inverse limit space (16 of the inverse system just constructed, where B indicates that such a limit space exists for each basis, B, of X. We note that by Theorem H, X°(B) # ¢. We define a map f:X 4 Xa(B) as follows. We give each coordinate of images using the canonical projection Tra:Xon 4 Xa for each a E I. For each x e X and a E I let wa(f(x)) = fa(x). That the resulting map f is well defined and continuous follows from Theorem B and (l) of Theorem J. That is, f is the induced or limit map from X==lim{X,:i into X ==lim{X ,f I oB]I B I where the inverse system of maps is (fa: a g I}, since Wa° f = fa and the limit map is unique with respect to this property. Theorem ;: The map f:X 4 f(X) is a continuous open sur- jection. Proof: Continuity is shown above and the surjective property is obvious. To show that f is open it is suffic- ient to show that for each U e B, f(U) is open in f(X). Let a = {U} c B. From the proof of (ii) in Theorem 1, fo(U) = Uo is open in X. Since Hg is continuous, w;l(Ua) is open in Xan and then w;1(Ua) n f(X) is open in f(X). Claim that f(U) is w;1(ua) n f(X). This is quite clear when from (iii) Lemma 1 we see that Ua is l4 indeed a singleton in Xo and that f;l(Ud) = U. Further, from the discussion preceding this theorem, -1 _ -1 nu (0a) m f(X) _ f(fa (van. CHAPTER THREE FINITE RESOLUTIONS: EXISTENCE AND GENERAL PROPERTIES In this chapter we see how the constructions of the pre- vious chapter motivate the definition of a finite resolution of a To - space. We further find that these constructions encompass all finite resolutions. First we find how TO - separation of the given space affects the structure previous- ly given. Theorem ;: The mapping f:X 4 X“ of Theorem 2 is an imbed- ding iff X is a T0 - space. nggf: Since each factor space, Xa' is a To - space, f(X) is always a T0 - space. Hence if map f is an imbed- ding then X is a To - space. Conversely, if X is a To - space then we need only show that the map is injective and the result follows by Theorem 2. Let x,y e X be distinct points. In basis, B, there is some open set U such that either x e U and y E U or x f U and y 6 U. In either case, letting a = [U] E I we find by the definition of f:X 4 Xm that f(x) # f(y) since their a-—th coordinates fa(x) and fa(Y) are dis- tinct. Hence the map f is an imbedding. dB]I' Subset F c X0° is strongly dense in X iff for each a e I and xa E Xa' Definition: Let X = lim[Xa, f G) 15 l6 '1 Fa (Xa) 0 F ¢ ¢- We note that strongly dense implies dense in the usual sense for a subspace because of Theorem F. It also implies that each Va is surjective. Henceforth when X is a TO- space and imbedded in an inverse limit space as in Theorem 3 we will write X c Xm, identifying X with its image. By the definitions in the constructions we have seen that v o f = fa in the above. Then considering X c Xan ‘we have a. f = w and where confusion is not likely, we will write a aIX w :X 4 X . C1 (I f and let X be a oB}I To - space. Then X0" is a finite resolution of X iff Definition: Let X = lim{X , a: G- a (i) Each Xa is a finite To - space (ii) X can be imbedded as a strongly dense subspace of X . on Of course the constructions motivate this definition and it is clear that the inverse limit space of Theorem 3 meets these requirements. The strong density follows from the non~ void condition on the subsets of a partition. That is, each xa e Xa corresponds to a nonvoid subset of X and so con- sidering X c Xco after imbedding, w;1(xa) n X = xa c X. We temporarily call such a finite resolution and any natur- ally equivalent to it a basic finite resolution, since it is constructed from a basis for X. However we now prove the following. 17 Theorem 3; Every finite resolution of a To - space is basic. Proof: Let X = lim[X , f I be a finite resolution _""—""‘ on Q— a (161 of T0 - space X. By Theorem F we construct basis B = U G _ QEI for X where G = {w 1(U ) n X: U open in X I for each a o a a o a 6 I. We must now show the following. (i) That Xo has the same number of points as the finite space generated by Go via the relation (Go) as described in Lemma 1. (ii) That walx: X 4 Xa is an identification map. (iii) That the collection {Ga} is cofinal in the set of all finite subsets of B, ordered by inclusion. Then by Theorem D, X0° is naturally equivalent to the basic finite resolution constructed from B. (i) Since Xo is a T0 - space we have for each x E X r1 (1 that {xa}=n[Ua e wama): xa e um} - UIUa e Wa(Ga): xanaI Since inverse maps commute with intersection, union and com- plementation, the characterization of points in the finite space generated by relation (Go) given by (iii) of Lemma 1 shows that the partition associated with (Go) is identi— cal to that of w;l(Xa) n X. (ii) Since X is strongly dense in Xm the map na:X 4 X” is surjective. The surjective restriction of an identifica- tion is again an identification. The map wa:X 4 Xo is the restriction of Wu: H)&14 Xo which is an identification map . I. ll, \Ilu. III III I ll 18 (in fact an open mapping). (iii) Let u be any finite subset of B. For each U 6 u choose some a E I such that U 6 Go' Label the resulting finite subset Io c I. Since I is a directed set there is some B E I so that a s B for each a E 10' Hence it is sufficient to show that for o g 6, Go c GB. Let Uo be open in Xo' Then f;;(Ua) = VB is open in X6 and v-1(v ) n X E G . By Theorem B we get B B B w“1<"°>
    ‘Xe‘B’I 4 XQ(B) BBI such that i is the identity map on X. If we then write BB' X B’ = 1' then X (B) is homeomorphic to D 25 II where I = [o c B c B’: o e 1’} c I’. That is just the natural projection given by Theorem limIXa, foB is, QBB’ D from lim[Xa, faBII, into lim [Xa' faBII. Then [Xm(B),BB,I is an inverse system of spaces. By (2) of Theorem J the resulting inverse limit space is homeomorphic to X(T) where T is the basis of all open sets of X. Hence inverse lim- its of finite resolutions yield nothing more of interest. We call X(T) the total resolution of X. Homeomorphic finite resolutions occur for certain col- lections of bases as shown in the following. Lemma ;: Let a be a finite nonvoid collection of open sets of space X. Let a + = [intersections of sets from d}. Then the finite TO - spaces, Xo and Xo+' generated by a and o+ are homeomorphic. Proof: It is sufficient to show that the partitions Xo and Xa+ generated by relations (a) and (a+) are identical. Since a : a+, properly refines Xa and so for x,y E X x(a+)y implies x(a)y. Conversely suppose that x(a)y. Let ux = [U E o+= x 6 UI and uy==[U E a+: y 6 UI. We wish to show that ux = uy which implies that x(a+)y. Hence x(a)y iff x(a+)y and the partitions are iden~ tical. Case 1: ux = ¢. If y E U E o+ then there is some V E o with y e V as U is an intersection from d. Then x(o)y implies that U E ux. This contradicts ux = ¢ and hence uy = ¢. 26 Case 2: Let x E U 6 a+. We may write U = nUi where each Ui E Q. But x E U iff x E Ui E o for each i iff y E Ui 6 o for each i iff y 6 U. Hence ux = uy. Theorem p: Given a basis B for a T0 - space X, construct basis B+ = [finite intersections from BI. Then finite reso- lutions Xw(B) and X”(B+) are homeomorphic. Proof: First note that if I: [ocB: a74¢ and finite} then I+= [o+§:B+:a<;BI is cofinal in J: [y:B+:y;é¢ and finite}. Hence by Lemma 2 and Theorem C the finite resolutions above are homeomorphic. The homeomorphism intended here is lim[maI = ¢:Xm(B+) 4 Xm(B) where 4 Xo is the homeo— mazxa+ morphism of Lemma 2 that preserves subsets of the partitions. Corollary: Let B’ be any basis such that B c B’ c B+. Then Xm(B’) and Xm(B) are homeomorphic. Proof: The induced homeomorphism of Theorem 8 is clear- ly the same as the projection of Example 1 where B c B+. Hence any B’ in the position given yields a finite resolu- tion homeomorphic to Xm(B) as it is projected through this homeomorphism. More specifically, for a c B, (a+} is co- final in [B c B+: B # ¢ and finite} and this implies that [a+ n B’: a c B} is cofinal in [y c B’: y # ¢ and finite} and the generated finite spaces are still homeomorphic to the original Xa's. CHAPTER FOUR COMPACT HAUSDORFF SPACES In this chapter we find that separation properties and then compactness for a given topological space force a more rigid positioning of this space in its finite resolutions. Several characterizations and properties are found in the compact Hausdorff case. Definition: Let S be a topological space. Subspace X c S is T2 - separated in S iff for distinct x,y E X there are disjoint open sets U,V in S such that x E U and y 6 V. Subspaces X and Y of S are T - separated 2 in 8 iff there are disjoint open sets U,V in S such that X c U and Y g V. Lemma 3; Let XQ(B) = lim[Xa, be a finite resolution f QB}I of Hausdorff space X. Then X is T2 - separated in Xco Further, if Y and Z are T2 - separated in IX then ‘Y and Z are T2 - separated in Xco Proof: Since X is Hausdorff there are U,V 6 B with U n v = ¢. x e U and y e v. Let a = [U,VI e I. Then by (iii) of Lemma 1 w (U) = U and v (V) = V are disjoint o a o a open sets in Xo' Continuity of Va implies that h;1(Ua) —1 and fig (Va) are disjoint open sets in XOD containing x and y respectively. Now let u and U be disjoint open sets of X with ‘Y c u and Z c U- Then u = UU and 27 28 U = UV where U's and V's are basic open sets from B. For each U involved in UU let GU = {U} 6 I. Then the factor space X has two points x and y where 1 ou 0U oU w- (x ) n X = U. Then clearly "U “U -1 [U ( ) nx=u. w x ] Similarly define av and x0 and get V [U W-1(X )] n X - U “v 0v Since singletons x and x are open in X and X _1°‘U 0’v _1 GU 0v the sets u’ 2 UN (x ) and U’ = U W (x ) are open in 0U “U X” and they contain u and U and hence Y and Z re- spectively. If u’ n U’ # B then since X is dense in X”. u’ n U' n X # ¢. But this contradicts u n U z ¢ since by construction we have u’ n u’ n X = u n U. So U' 0 U’==¢ and Y and Z are T2 - separated subspaces in X” We note that a similar definition and result is easily) proved for T1 - spaces. Also, the condition of being a T2- separated subspace is a strictly stronger condition than that of being a Hausdorff subspace. The importance of this is seen in the following. Definition: Let S be a topological space. Subspace X c S is co-dense in S iff for each s e S, s n X # ¢. The closure of course is taken in S. Example ;: Let S = [x,y,z} with topology given by T = [¢,S,[x,y},[y,z},[y}}. Then X = [x,z} is co-dense in S. \III III I] ‘IJ III 29 Clearly, for T1 - space S, X is co-dense in 8 iff X = S. HOwever, Example 2 shows that X being co-dense in S does not imply that X is dense in S. Further, note that in Example 2 the subspace X is Hausdorff but not T - 2 separated in S and [y] = S. This shows that T2-separation is necessary in the following. Lemma g; Let X be co-dense and T - separated in S. Then 2 §.n X is a singleton for each s 6 S. Proof: Let s e S and let x,y 6 X be distinct points. By T - separation there are disjoint U and V open in 2 S with x e U and y e V. Hence at most one of the ele- ments x and y can be contained in {s}. We now prove a general result which is used in charac- terizing the position of a compact Hausdorff space in its finite resolutions. Lemma 5: Let X” = lim[Xa, f be an inverse limit of dB}I finite spaces. Then Xm is compact. Proof: Let Xé be the same point set as Xa' but with the discrete topology. (Then for each a E I the correspond- I . . . . . ence xa * X defines a continuous bijection mo:xd 4 X a a' Then if X’ = lim[X’, f’ ‘where (x’ = x’ iff a e a I dB]I foB B a ’ = ’ ' ' inverse foB(mB(xBn ma(xa), the collection {ma} is an system of maps with limit map §:X; 4 X0° also a continuous bijection. Since Xé is compact and Hausdorff, x; is com- pact and Hausdorff by Theorems G & H. Then Xco is compact as the image of x; via Q, 3O Corollary: The inverse limit space of an inverse sequence of finite spaces is the continuous image of the Cantor dis— continuum. Proof: This is discussed further in Example D of chap- ter five. Corollapy: Every finite resolution of a T0 - space is com- pact. Every finite resolution of a connected To - space is a T0 - continuum. Proof: The first result follows directlwarom the previous lemma. The second follows from the additional ob— servation that the space is dense in its finite resolution. The following is a well known result on compact To - spaces. Lemma 6: Let S be a compact To - space and let F be any nonvoid closed sUbset in S. Then F contains a closed singleton. 3392;: Let 3' = [Fi e F:Fi 7! o and F1 closed in F}. Since F is closed in S, each F1 is closed in S. Or- der J'by inclusion and let a. be any simple chain from J. By the compactness of S, n c.# ¢ and so 0 c,e J. By Zorn's Lemma, J has a minimal element M # ¢. Let x,y E M. The minimal property of M implies that §'= §'= M. Since S is a T0 - space this implies that x = y and hence M is a singleton. 31 The next result is a direct consequence of Lemma 6 but to the best of our knowledge it is not well known. Ibeprem g: Let S be a compact TO - space and let X be any subset of S such that every closed singleton of S is contained in X. Then X is compact. Proof: Let u be any open covering of X. Then u also covers S. If not then S - U1; is nonvoid and closed and must contain a point in X. Contradiction! Now 8 is compact and so u contains a finite subcovering of S and hence of X c 8. Therefore X is compact. Theorem 10: Let XOD = lim [Xa, be a finite resolution f oB}I of Hausdorff space X. The following are equivalent. (i) X is compact. (ii) X is co-dense in X“. (iii) X = [x e Xeo : [x] = {XI}. 2399;: (i) implies (ii). By the strong density of X in Xm. J = [U;l(;;727) n X =a.E I} is a collection of non- void closed subsets of X for each 2 E X. Since I is directed, (ii) of Lemma 1 implies that J is a filterbase on X. X is compact and so my 5£ Q5. By Theorem I, nJ= E n x. (ii) implies (iii). By Lemma 3, X is T2 - separated in X00 and so by Lemma 4, [x e Xm::Tx_ = [XI] c X. Now let x E X and y e X with y # x. If y E X then Lemma 3 im- plies y f E; If y Z X then X being co-dense implies that there exists 2 e X with z E y. Suppose y e E1 32 Then 2 6‘; and so 2 = x. Hence x 6‘; and y 6‘; and this contradicts the To - separation of X”. 80 y f El We have shown that for each y # x, y z i: That is, 2': x and X; [x exm:_[;I-= {XI}. (iii) implies (i). Direct from Theorem 9. We note that the equivalence of (ii) and (iii) is more general than given above. The structure of Xon is not necessary, but just that X is a T2 — separated subspace of a compact To - space. That is sufficient for (ii) implies (iii) and then (iii) implies (ii) by Lemma 6. Corollary: Let Xco be a finite resolution of a compact Hausdorff space X. Then X is a maximal Hausdorff sub- space of Xm. Proof: Let X g Y ; XOD such that Y is Hausdorff. If y E Y -X then there is some x e X such x 6 §; since X is co-dense in X . Hence Y is not T2 - separated in (D XOD but Xm is a finite resolution of Y a X. This contra- dicts Lemma 3 and so Y -— X z ¢. A final indication of the position of X c X in the co compact Hausdorff case is given in the following. Theorem 11: Let Xm(B) = lim [Xo' f be a finite resolu- dB}I tion of a compact Hausdorff space X. Then X is a retract of X o m Proof: Define r:Xon 4 X as follows. For each 2 G X , let r(z) ='; n X. This map is well defined by Lemmas 3 a 4 33 and Theorem 10. That is, EDn X is a singleton. Also, by Theorem 10, r(x) = x for each x e X c X”. To show that r is continuous it is sufficient to show that r-1(U) is open in X0° for each U 6 B. Let 2 E Xco such that r(z) E U. Since X is normal there is V E B with r(z) e V c'V c U. Let a = {V} E I. We have seen that in Xo the set wa(V) = Va is an open singleton. Then w;l(Vd) is open in Xm. By this construction wa(r(z))==va. Hence w (z) = V since if v (z) = X - V then a o a o a wa(r(z)) = wa(z n X) ; TTa(Z) C Wa(Z) = Xa— Va contradicting the construction. So 2 e W; (Va). It re- mains to show that w;l(va) : r—1(U). Suppose not and y E w;1(va) such that r(y) E U. Then the normality of X implies the existence of W E B such that W DIV = B and r(y) e w. Now let [3 = {v,w} e I. In x9, 7TB (V) = VB and WB(W) = WB are open points and Va and VB represent the same subset in the partitions of X. We have the following contradiction. W = w B B(r(y)) =WB(ynX) swam CTrBW): VBsXB “-WB. l -1 Hence 2 E w;l(va) c r- (U) and so r (U) is open in Xm Corollagy: If X is a compact metric space then Xon is a semimetric space. 34 Proof: Let d:X) 1 where the fij are the induced bonding maps. Hence {mi} is an inverse system of maps with continuous limit map é: Y 4 X = lim [X., f..} m e i 1] by (l) of Theorem J and Theorem A. Now let r:Xco 4 [0,1] be the retraction of Theorem 11. Then let m = r o i. By the definition of r and Theorem I, if y E A then mm = rem) n [wilhkmw n x] k—l m -l m 1 =fl[O'—']‘("]:O I—' M 42 Similarly, o(y) = l for each y e B. In general a finite space X with an odd number of points, xl,x2,° will be called a 2n+l - train or ..'x2n+l’ simply a train iff X is homeomorphic to the given set of points with subbasis k = 0,l,"',n+l}. Hx2k-1' x2k'*2k+1}‘ Example p: Let X = [0,1] with the upper limit topol— ogy. Let B be the dyadic intervals of Example C with the right end points included. Now the partitioning of X oc— curs as a collection of open sets and so Xk has 2k points and is discrete. Further since each point xk e Xk has ex- actly two predecessors in level k+l, the resulting limit space Xon is homeomorphic to the Cantor discontinuum. See [4] pp. 97-100. In this case threads which are eventually right directed represent the dyadic rationals with the excep- tion of 0 e X which is eventually left directed in X0° All other eventually left directed threads make up Xm-X. We note that by Lemma 5 the Cantor discontinuum is a Hausdorff compactification of [0,1] with the upper limit topology. Example g: Let X = [0,1] with basis B consisting of the dyadic intervals of Example C and the sets of dyadic rationals contained in each of these intervals. Each level space Xk is generated by the dyadic intervals with endpohum; n/2k for n = 0,l,"',2k and those sets of dyadic rationals contained in these intervals. From Example C, Xk has 3-2k - 1 points. For example, Xl has the following sub- sets of X as its points. 43 {dyadics in [0,1/2)} X1.1 = x12 = [0,1/2) - x11 X13 = [1/2] xl4 = {dyadics in (l/2,l]} x15 = (l/2,l] —- xl4 Basic open sets in X1 are [x11],{x14I,[x11,x12I,{x14,x15}. has two predecessors in X2 and each of these etc. These points represent Clearly, x12 has two predecessors in X3, partitioning of the nondyadic reals in [0,1/2) into those in [0,1/4) and (l/4,l/2), etc. In each level these points are closed singletons by our construction. Hence for -l a 6 7T1 (X12)! {a} is a closed singleton by Theorem I. And since for each such a g Xco that is eventually right (left) directed, BIN-10:1 ) n X] = ¢ k k ’ we must have a E A where A is the subset of Xco defined in Lemma 8. Now X is Hausdorff but not regular and so X U A, be- ing compact, cannot be Hausdorff since every subspace, in- cluding X, must then be completely regular. What has oc- curred in this case is that some point of X is not a closed singleton in X“. Such a point is O E X. In coordinate 0‘— form we write 0 = (xkl) and see that (xkz) E {0} since sz E {Xkl} for each R = 1.2.'°'. See Theorem I. 44 Example F: Consider the following plane continuum, X = U In' homeomorphic to the topological sine curve. n=0 17 ,f 1 ft I //\ fl"! I I I / \~. I I 11/, I I / 12 \I’ 0 I/ I oI’ 1/2 3/4 1 Diagram F At step k, each interval In is partitioned as [0,1] in Example C. To insure finiteness, Xk is generated by the usual open sets in In for n = l,2,"°,k. with the ex- ception that the only included endpoint occurs at (0,0) and the open sets formed by "slicing" U I-{(1--%E,JJI In n=k+1 k2 with horizontal lines through y-coordinates j/2k for j = 0,1, - - -,2k. The collection of all these open sets through steps k = l,2,°°° is a basis for X and the collections at each step form a cofinal sequence for the set of finite subsets from this basis. At each step k, Xk has (k+1)2k+1- points and is a train as described after Example C. In fact X in this example is a train of the same length as 2P—1 X2P+p-1 in Example C. This means that the unit interval and the topological sine curve have resolutions involving the same sequence of finite spaces. Of course the bonding maps are considerably different as in the present example a "fold- ing" of the X 1 train onto the Xk train occurs. k+ 45 Example p; Let X be the unit circle. We continue to form dyadic type nested partitions generated by nested col- lections of open sets. In this case let X have its usual position in the plane with center at the origin. The open sets generating Xk are all open arcs with endpoints on the radial lines 9 = 2n'n/2k radians for n = l,2,-:-,2 . For k 2 2, finite space X is essentially a 2k+l + 1 train k with its tails identified. So Xk has 2k+1 points and we call it a 2k+l - block. A tree representing the result- ing Xco appears in Diagram G. The three branches from the root represent the upper and lower open semicircles on the left and right with a dyadic subdivision as in Example C. The central branch contains only the two threads represent- ing (1,0) and (-l,0) on the unit circle. Diagram G Example H: Let X be an annulus. By Theorem 6, the inverse limit of spaces Xk which are products of 2k+l - 1 trains and 2k+2 blocks is a finite resolution of the annu- lus, which is the product of an interval and a circle. Then Xk has 2k+2(2k+l - 1) points for k = l,2,°°'. We illu- strate the subdivision X1 and a schematic of this space as 46 three nested 8-blocks in Diagram H. The partitioning of X continues both radially as in Example F and concentrically in X2, etc. We note that the resulting inverse limit space is also a finite resolution for the open annulus as this subspace is strongly dense. Again by Lemma 8, X is the only compactification of the open annulus present in X” Diagram H CHAPTER SIX ALGEBRAIC STRUCTURES In this chapter, X is always a compact Hausdorffspace. We find then that some of the common algebraic structures re— lating to X are inherited by the finite resolutions of X. Theorem lg: Let X“ be a finite resolution of ){ and let ¢:Xm 4 Xco be a homeomorphism. Then é/X is a homeomorphism from X onto X. Proof: By (iii) of Theorem 10, X is exactly the sub- set of closed singletons in X”. i and Q are closed maps so ¢(X) ; X and ¢-1(X) : X Applying e to the latter inclusion shows that X c ¢(X) and hence @(X) = X. Corollary: X has the fixed point property iff X0° has the fixed point property. Corollapy: Let H(Xm) and H(X) be the homeomorphisnugroups of X” and X respectively. Then the map H‘HIXm) 4 H(X) given by “(4) = é/X is a homomorphism. Lemma 2: Let m:Y 4 Y’ and @:Z 4 2’ be homeomorphismsanui let p:Z 4 Y and q:Z’ 4 Y’ be continuous surjections such that the following diagram commutes. 47 Then for each y e Y and y’ 6 Y’, p-1(y) and q-l(m(y)) are homeomorphic and q-1(y’) and p-1(m-l(y’)) are homeo- morphic. Proof: Simply by the commutativity. The homeomorphisms readily available are the restrictions of i and 4-1 to P (Y) and q—l(y’) respectively. Lemma IQ: Let Xco be a finite resolution of X and let map- ping r:Xco 4 X be the retraction of Theorem 11. Let m:X-+X be any homeomorphism that has an extension :X0° 4 X“. Then r-1(m(x)) = 4(r—1(x)) for all x E X. ggppf; With r taking the place of p and q in Lem- ma 9, we have the required commuting diagram since the clos- ure operator commutes with homeomorphisms. For example, if z e X... co’IIerm = cp'lde')‘ n X) = so—IUIE) n X) = WM?) 0 x) = 2.0 X = r(z) Then r-1(x) and r_1(m(x)) are homeomorphic via e, which in this case means ¢(r-1(x)) = r-1(m(x)). 49 We note that by Theorem 12 we have also shown that every homeomorphism @:Xm 4 Xco must send fiber r-1(x) into fiber r-1(¢(x)) for each x E X. If 2 e r-l(x) and z # x then say that z is a neighbor of x E X. Theorem I3: Let Xc(B) be a finite resolution of X. Let m:X 4 X be any homeomorphism such that B(B) = B. Then there exists a homeomorphism ¢:Xm 4 Xco extending m. Proof: This theorem is really a special case corollary to Theorem 7, which guarantees the existence of Q as con- tinuous. That 6 is a homeomorphism follows from the con— struction of homeomorphisms ma in Theorem 7, w(B) ==E3 and Theorem C. The necessity of w(B) = B is easily seen in light of Lemma 10 as x e X and its image may have fibers of differ- ing cardinality in case w(B) ¢ B and so no extension is possible. As an example, let X = [0,1] and let B be 1 the usual open interval basis and B is the dyadic inter— 2 val basis of Example C. Define a new basis B as the union of the collections [0.3/4) n B and (1/4,1] n B 1 2 Consider the homeomorphism ¢(x) = l-x. Now let x < 1/4 be other than a dyadic rational point. Since the collection [(x,l/n) = n = 2,3,---} g B , the construction of points in Xco implies that r-1(x) is not a singleton. Since m(x) > 3/4 and not dyadic, the results in Example C imply r—1(m(xn = m(x). 50 gorollagy: Let Xw(T) be the total resolution of X. Then H(X) is a quotient of H(Xm). Egggg: For every m E H(X) ‘we have TIT) = 7, since- T and m_1 are open mappings. By Theorem 13 every m E H(X) extends to some i E H(Xm). Hence the homomorphism of the second corollary to Theorem 12 is surjective. Of course there are examples where a basis courser than T will suffice in the above. For instance, let X = [0,1] and let B = [open intervals} or more generally, let X be locally connected and let B = {locally connected open sets}. The following results discuss the case of arbitrary basis and examine some groups of homeomorphisms as subgroups of H(X) and H(Xm). Definition: Let H(X) and H(Xm) be the homeomorphism groups of X and XQ(B) respectively. We then write H(XoB) = {m E H(X) = m(B) = B] H(Xm,B) = [e e H(Xm) : @/x e H(X,B)} H0(Xm,B) = [e 6 H(XQ,B): i is the extension of Theorems 7 e 13} We note that Hb(Xw,B) is well defined by Theorem B and is a subgroup of H(Xm,B) by Theorem A. Simple examina— tion then proves the following. Lemma ll; The following inclusions are as subgroups. (i) H(XaB) C H(X) (ii) H(XQ,B) g; H(Xm) 51 (iii) HO(XQ,B) ; H(Xm,B) (iV) H(X,Bl) n H(X.B2) s H(X.Bl U 32) Theorem lg: Ho(Xm,B) and H(X,B) are isomorphic. Proof: By Theorem B the correspondence lim mo m is one-to-one, where the ma are as in Theorem 7. Hence the correspondence is an isomorphism by Theorem A. In the following, Hb(X) and Hp(Xm) are the p-th Cech homology groups of X and Xco respectively. The nota- tion is essentially the same as in [4}. The underlying ne- cessary group structure is not included in the notation. we let XXX) be the collection of all finite open coverings of X. For a,B E ZXX) write a < B if B refines o. For finite a,B c B, a given basis for X, write a S B if B contains a. Let I = [finite a c B}. The compactness of X implies that the collection, J, of basic finite open coverings from B is cofinal for both. (ZNX),<) and (1,3). Unfortunately these orderings are not compactible and so we proceed further by defining for each a 6 J and x E X, the open set UaIXI =n (Ugh: erI. The finiteness of o also implies that the covering ha = [Ua(x): x E X} is finite for each a E J. By the construction, a < ua for each a E J and so [ua: a 6 J} is cofinal in (XXX), <). 52 Further a < B implies that ua < u since B contains a B and xEX means that UB(x) =n [UeBszU};n{UEa:x€U]. Now let Hp(u) be the p-th Cech homology group of the nerve of u E XXX) considered as a simplicial complex. For any n > u let W : U a u be the usual projection and de- UU note the resulting induced homomorphism by *wuu_ from V Hp(u) to Bp(u). The p-th Cech homology group of X is the inverse limit group Hp(X) = 12m {Hp(u). ]. *Wuu Let X (B) = lim {X , f } be a finite resolution of a: O— 0. (I X. Let Hb(xa) be the p-th Cech homology group of the fac— tor space Xa. Each bonding map induces a homomorphism writ— ten f : H X *aB p