AFOSR-798 THE U N IVE R S I T Y OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Note REGULAR CONVERGENCE IN A PARACOMPACT SPACE J. H. Stoddard ORA Project 03597 under contract with: MATHEMATICAL SCIENCES DIRECTORATE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH CONTRACT NO. AF 49(638)-774 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1961

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1961.

TABLE OF CONTENTS Page ABSTRACT v INTRODUCTION 1 CHAPTER I. CONVERGENCE OF NETS 7 CHAPTER II. COHOMOLOGICAL N-REGULAR CONVERGENCE OF NETS 11 CHAPTER III. CONVERGENCE AND GENERALIZED MANIFOLDS 26 BIBLIOGRAPHY 34 DISTRIBUTION LIST 35 iii

ABSTRACT The purpose of this thesis is to systematize the main results concerning n-regular convergence using a uniform method and placing them in the framework of a general, possibly non-metrizable, paracompact topological space. Let X denote a fixed paracompact topological space and C(X) the family of non-empty closed subsets of X. For a collection u of open subsets of X, let N(u) denote those elements of C(X) that have a non-empty intersection with each U c u and are contained in the union of all U c u. We give C(X) the topology generated by sets of the form N(u) where u is a locally finite collection of open subsets of X. We say a net (Ai) in C(X) converges to A if it converges in this topology. By a closed covering of X we mean a locally finite collection of e of closed subsets of X such that each x e X is in the interior of some E e e. For a closed subset F of X, we let Hq(F) V denote the q-dimensional reduced Cech cohomology group of F with coefficients in a principal ideal domain L. A net (Ai) in C(X) converges cohomologically n-regularly to A if (Ai) converges toA and for each closed covering e of X there exists a closed covering d of X, a function E from d into e, and a j e I such that D C E D for each D e d and the natural homomorphism of Hq(n DyrAi) into Hq(DrAi) is trivial for all q c n and all i > j. We use the abbreviation n-rc converges for this type of convergence. We indicate a few of the results proven in this thesis. If (Ai) is a net in C(X) n-rc converging to A, there exists a j E I such that Hn(Ai) v

and Hn(A) are isomorphic for all i > jo If (Ai) is a net in C(X) (n-l)-rc converging to a non-degenerate set A, each Ai is a connected n-dimensional generalized manifold over a field, and each x e A has an open neighborhood W such that WCAAi is orientable for all i e I, then A is a connected n-dimensional generalized manifold over the field. If (Ai) is a net in C(X) (n-l)-rc convergingto a non-degenerate set A, each Ai is a connected compact orientable n-dimensional generalized manifold over a field, then A is a connected compact orientable n-dimensional generalized manifold over the fieldo If (Ai) is a net in C(X) (n-l)-rc convergingto a non-degenerate set A where each Ai is a connected compact orientable generalized n-manifold with boundary Bi over a field and (Bi) (n-2)-rc converges to a nondegenerate set B, then A is a connected compact orientable generalized nmanifold with boundary B over the field. For a regular topological space, every open covering has an open star refinement if and only if the space is paracompact. To the extent this covering property is required for investigating n-rc convergence, a paracompact topological space providesa natural setting for the theory. vi

INTRODUCTION The convergence concept pervades most of mathematics. For example, much of analysis is concerned with sequences of functions converging to a limit function. Under appropriately strong forms of convergence, the limit function inherits certain properties the members of the sequence may possess. Similar questions arise in topology concerning sequences of subsets of a topological space. Let X be a compact Hausdorff topological space. For a sequence (Ai) where each A. is a subset of X, let lim sup A. denote those x E X with the property that each neighborhood of x contains points from infinitely many of the Ai. Let lim inf Ai denote those x e X with the property that each neighborhood of x contains points from all but a finite number of the Ai. If lim sup Ai = lim inf Ai, the common set A is the limit of the sequence (Ai). Since lim sup Ai is always a closed subset of X, the limit set A is always closed. Also, a sequence of subsets of X converges to the limit set A if and only if the sequence consisting of the closures of the members converges to A. For this reason we consider only sequences whose members are closed subsets. This definition of convergence was introduced by Zarankiewicz [16]. It follows easily that, if (Ai) converges to A where each Ai is a closed connected subset of X, then A is a closed connected subset of X. However, a stronger type of convergence is required -1

-2to insure the limit set A will be, for example, a simple closed curve if each Ai is a simple closed curve. G. T. Whyburn [13] introduced the concept of a sequence (Ai) of closed subsets of a compact metric space X converging regularly to a subset A of Xo He required that the sequence (Ai) converge in the above sense and that for each positive number e there exist a positive number d and an integer j such that two points of A. at distance less than d are contained in a connected subset of Ai with diameter less than e for all i larger than j. Notice that in the special case where each Ai equals A, the above definition requires that A be uniformly locally connected. He showed, among other things, that the limit set A is always locally connected and that the simple closed curve property is inherited by the limit set A. G. T. Whyburn [13], also, introduced the concept of a sequence (Ai) of closed subsets of a compact metric space X converging n-regularly to a subset A of Xo He used the Vietoris homology theory in formulating his definition and showed that 0-regular convergence corresponds to the above regular convergence. Over a period of years several authors investigated this situation. P. A. White [12] contains a summary of the results obtained. For example, Eo Go Begle [1] obtained the following extension of the above result concerning simple closed curvesIf a sequence (Ai) of closed orientable classical 2-manifolds

-3converges 1-regularly to a non-degenerate set A, then A is a closed orientable classical 2-manifold and A is homeomorphic with all but a finite number of the Ai. The lack of a topological characterization of classical n-manifolds with n larger than 2 has prevented further progress in this directiono However, using the generalized manifold introduced by Ro L. Wilder [14], E. Go Begle [1] obtained the following result, If a sequence (Ai) of compact orientable generalized n-manifolds converges (n-l)-regularly to a n-dimensional set A, then A is a compact orientable generalized n-manifoldo Go So Young, Jro [15] investigated n-regular convergence of a sequence (Ai) of closed subsets of a locally compact complete metric space X. He extended the above 2-dimensional result of Eo G. Begle to non-closed classical 2-manifolds with boundary. Po A. White [10] considered the problem of extending the regular convergence theory to sequences (A,) of closed subsets of a possibly non-metric, compact Hausdorff topological space. However, his definition of regular convergence differs from the classical definition. He fails to relate his definition to the classical one because of an error in the proof of [10, Theorem. 10]o In proving [10, Theorem. 10] he uses [10, Corollary 7.2] to show that the classical definition implies the definition introduced earlier in his paper. However, the

-4hypothesis of this corollary requires that the sequence converge in this earlier sense. E. E. Floyd [5] defined n-regular convergence for a sequence (Ai) of closed subsets of a possibly non-metric, compact v Hausdorff topological space Xo He used the Cech homology theory with coefficients in a field or compact group and established the following result: If a sequence (Ai) of closed subsets of a compact Hausdorff topological space X converges n-regularly to A, then the v n-dimensional Cech homology group of Ai is isomorphic with the n-dimensional Cech homology group of A for all but a finite number of the Aio It is our intention to systematize the main results concerning n-regular convergence using a uniform method and placing them in the framework of a general, possibly non-metrizable, paracompact topological space. In a certain sense, which we will make precise later, this is a natural setting for the n-regular convergence theory. Let X be a paracompact topological space. Let C(X) denote the family of non-empty closed subsets of Xo For a collection u of open subsets of X, let N(u) denote those elements of C(X) that have a non-empty intersection with each U c u and are contained in the union of all U E u. We give C(X) the topology generated by sets of the form N(u) where u is a locally finite collection of open subsets of X. Since X may not be a

-5metric space, it is natural to consider nets in C(X) in the sense of J. L. Kelley [6] rather than sequences in C(X). We say a net (Ai) in C(X) converges to A if it converges in the above topology. By a closed covering of X we mean a locally finite collection e of closed subsets of X such that each point x E X is in the interior of some E e e. For a closed subset F of X, we let Hq(F) denote the qdimensional reduced 6ech cohomology group of F with coefficients in a principal ideal domain L. If e and d are closed coverings of X, we write d > e if there exists a function rr from d into e with DC 7rD for each D e d and such that the natural homomorphism of Hq(T-D) into Hq(D) is trivial for all qn. If e is a closed covering of X and F is a closed subset of X, we let e c F denote the closed covering of F consisting of elements of the form ErF where E e e. A net (A.) in C(X) converges cohomologically n-regularly to A if (Ai) converges to A in C(X) and for each closed covering e of X there exists a closed covering d of X and j from the index set I for the net such that drNAi > erNAi for all i larger than j. We use the abbreviation n-rc converges for this type of convergence. We indicate a few of the results proven herein. If (Ai) is a net in C(X) n-rc converging to A, there exists a j e I such that Hn(Ai) and Hn(A) are isomorphic for all i larger than jo If (Ai) is a net in C(X) (n-l)-rc converging to a non-degenerate set A and each Ai is a connected compact orientable n-dimensional generalized manifold over a field, then A is a connected compact orientable n-dimensional generalized manifold over the field.

-6If (Ai) is a net in C(X) (n-l)-rc converging to a non-degenerate set A, each Ai is a connected n-dimensional generalized manifold over a field, and each x e A has an open neighborhood W such that WrAi is orientable for all i e I, then A is a connected n-dimensional generalized manifold over the field.

CHAPTER I 1,0 We consider convergence of nets in this chapter. Throughout this paper we let X denote a fixed topological space that is at least paracompact. If S is a subset of X, we denote the closure of S by S- and the interior by SO. By a nbdo of xE S we mean a subset N of X with x e N~ 1.1 Definition. A set I is a directed set if there is a reflexive, transitive, binary relation > defined on I with the additional property that for i, j e I there exists k c I with k > i and k >jo A net is a function whose domain is a directed set. A net is in a set Y provided its range is a subset of Y. If I is a directed set, we denote the net consisting of the pairs (i, yi) by (Yi)o A net (yi) in a topological space Y converges to y e Y if for each nbd. N of y there exists j e I such that yi e N for all i > j. 102 Definition. Let C(X) denote the collection of non-empty closed subsets of Xo For a collection u of open subsets of X, let N(u) denote those elements of C(X) that have a non-empty intersection with each U e u and are contained in the union of all U e uo We give C(X) the topology generated by sets of the form N(u) where u is a locally finite collection of open subsets of X. A net (As) in C(X) converges to A means the net converges in this topology. 1.3 Definition. If (A.) is a net in C(X) let lim inf A. denote the set of all x e X with the property that for each nbdo N of x there exists j e I such that Ai.NI0 if i >j Let lim sup Ai denote the set of all x e X with the property that for -7

-8each nbdo N of x and for each j e I there exists i e I such that i >j and AiriN0o. If lim inf Ai equals lim sup Ai, we write lim Ai=A where A is the common set. Since lim sup Ai is always closed, the limit set A is closed. 1.4 Proposition. Let (Ai) be a net in C(X). If (Ai) converges to A in C(X), then lim Ai=A. If X is compact and lim Ai=A, then (Ai) converges to A in C(X). Proof. Suppose (Ai) converges to A in C(X). Let x e A and let U be an open nbd. of xo Choose an open subset V of X such that A e N(u) where u = (U, V). Choose j e I so that Ai e N(u) for all i > j. Then AirUi0 for all i >j so that A lim inf Ai. Let x E lim sup Ai and suppose x ( Ao Choose an open nbd U of x with Ul-rA=0. Since X-U- is a nbdo of A in C(X), there exists j e I such that Ai c (X-U) for all i >j. Hence, AirU=0 for all i >j which contradicts the choice of x. Hence, x e A so that lim sup Ai CA, Hence, lim Ai=Ao Suppose lim Ai=A and that X is compact. Let N(u) be a nbdo of A in C(X) and let W denote the union of all U E Uo For each x E X-W, choose an open nbdo Ux of x and jx E I such that ArUx =0 and AriU =-0 for all i> jx Since X-W is compact, a x 1 x x finite number of the Ux's cover X-W. Choose j'c I such that it is larger than each of the jx's in the corresponding finite set. Since x X is compact, the number of elements in u will be finite. Hence, we may choose j" e I such that A.ir-UP0 for all U E u and all i > j"

-9Choose j e I such that j > j' and j>j" Then Ai eN(u) for all i >j. Hence, (Ai) converges to A in C(X)o Corollaryo If X is compact, lim satisfies the axioms of a convergence class for C(X) as given in J. L. Kelley [6]. Example. The following example shows that lim Ai=A does not imply convergence in C(X) in general. Let X be 2 2 the plane and let Ai be the graph of the equation x2 +i i= i where i is a positive integer. Then lim A. consists of the lines y=l, y= -1 while (Ai) does not converge to these lines in C(X). Lo Vietoris [9] showed, if X is compact, then C(X) is a compact Hausdorff topological spaceo E. Michael [7] observed the following proposition is valid 1. 5 Proposition. Let (A.) be a net in C(X) converging to A. If there exists j e I such that Ai is connected for all i> j, then A is connected. Proof. If A is not connected, A = BjC where B and C are non-empty open subsets of A with BrcC=9 o Choose open subsets U and V of X with UrA - B, VrA - C, and UV = 0. If u = (U, V), there exists j e I such that Ai e N(u) for all i j. Hence, all but a finite number of the Ai are not connected which contradicts our assumptions. Hence, A is connected. 1 6 Example. Let X be the plane and let A. be the set of points (x, y) with y=sin(l/x) and (1/i) e x' (l/7rr) for i a positive integer. Then (Ai) converges to the set A of points (x, y) with y=sin(l/x) and 0< x ~ (l/r) together with the points on the line segment between (0, 1) and (0, -1). Observe that each A. is locally connected, but A is not locally connected.

-10Remark. The above example shows a stronger form of convergence is required to insure, for example, that A will be a simple closed curve if each A. is a simple closed curve. 1

CHAPTER II 2o0 Let L be a fixed ring that we assume to be at least a principal ideal domain. We denote the q-dimensional reduced V Cech cohomology group of X with coefficients in L by Hq(X)o We V denote the q-dimensional Cech cohomology of X with compact supports and coefficients in L by hq(X). We recall, if U is an open subset of X with U compact, then hq(U) = Hq(X, X-U). If E and D are closed subsets of X with DCE, there is a natural homomorphism of Hq(E) into Hq(D). We denote the image of this homomorphism by IHq(E I D) and the kernel by KHq(E I D). If U and V are open subsets of X with VcU, there is a natural homomorphism of hq(V) into hq( U)o We denote the image of this homomorphism by Ihq(V U) and the kernel by Khq(V I U). 2.1 By a closed covering of X, we mean a locally finite collection e of closed subsets of X such that each x e X is in the interior of some E e e. If e and d are collections of closed subsets of X, d refines e and we write d >e if there exists a function rr from d into e such that DCTD for each D do If d > e and for each D d every element of d that meets D is contained in rTD, we write d > e. If d>e and for each D d, IHq(rD | D) = 0 for all q.n, we write dn>eo i o i io i i If d >e and for all D ~ d e H ( 1D s e ) = 0 D e d I D ~....nD for all q Sn, we write dn>>e. It follows easily that, if e, d, and c are collections of closed subsets of X with d > e and c nd, then dn>>e. Remark. If a is a collection of closed subsets of X, we let Hq(e) denote the q-dimensional reduced cohomology group of the nerve of e. If e is a closed covering of X, it follows easily from the -11

-12paracompactness of X that there exists an open covering u of X and ii i i a one to one function g of e into u such that g(E) contains E for each E e e and g(E o)r ~ng(E q)~ 0 if and only if E rn ~ ~o E q 0 o Since X is paracompact, open coverings of this type are cofinal in the family of open coverings of X. Hence, we may use closed coverings V of X to compute the Cech cohomology groups of Xo For a closed covering e of X, there is a natural homomorphism of Hq(e) into Hq(X)o We denote the image by IHq(e I X) and the kernel by KHq(e j X). In short, this notation always refers to the natural homomorphism from the group indicated on the left to the group indicated on the right. Eo Eo Floyd [5] proved a homology version of the following theorem with X compact and the coefficient group L a field or compact group. Using sheaf theory, Eo Dyer [4] proved a cohomology version with X compact and L a principal ideal domain, Recently, in a course at the University of Michigan, C. N. Lee used sheaf theory to prove the following version. 2.2 Theorem. Let E 19 E0, o, En be closed subsets of X with EjcE E forj=0, 1, ~, n. Let ej be a closed covering of Ej for j =-1,, 0 ~ ~, n such that en n>> ~ n e n>> e - Then 1. IH(E_ | En) C IH (e I En) for all q <n, 2 KIHq(e0 j E0) c KHq(e0 | en) for all q n +1. The results of this paper follow for the most part from applications of this theorem.

-132.3 Definition. X is q-clc if for each x e X and each closed nbd, E of x there exists a closed nbd, D of x with DCE such that IHq(E I D) = 0. If X is locally compact, X is 0-clc if and only if it is locally connected. If the coefficient domain L is a field, X is q-clc if and only if X is q-lc in the sense of R. L. Wilder [14]. X is clcn if X is q-clc for all q S n. 2.4 Definition. If e is a closed covering of a space X and A is a closed subset of X, let er-A denote the closed covering of A consisting of elements of the form EcA where E E e. 2.5 Definition. A net (A.) in C(X) converges cohomologically n-regularly to A if (A.) converges to A in C(X) and for each closed covering e of X there exists a closed covering d of X and j e I such that drA. n> eAi for all i >j. We use the abbreviation n-rc convergence for this type of convergence. 2.6 Proposition. Let X be compact and let (A.) be a net in C(X) converging to A. Suppose for each x e A and each nbd. E of x there exists a nbd. D of x with Dc E and j e I such that IHq(ErAi j DrAi) = 0 for all q n and all i ]j. Then (A.) n-rc converges to A. Proof. Let e be a closed covering of X. For x c E~ EE e, choose Dx and jx with the above properties as stated in the hypothesis. Since X is compact a finite number of the D's cover X. Let d be such a cover and choose j such that j > j for each j _~~~ x x corresponding to an element of d. Defining the function 7r from d into e in the obvious manner, we see that drtAin> erAi for all i > j Remark~ If X is a compact metric space and L is a field, n-rc convergence agrees with n-regular convergence defined in

-14terms of the Vietoris homology theory. If X is a compact space and L is a field, n-rc convergence agrees with the n-regular convergence v defined in terms of the Cech homology theory. 2.7 Proposition. Let (A.) be a net in C(X) n-rc converging to A. If e is a closed covering of X, there exists a closed covering d of X and j e I such that dnAi n> erAi for all i > j Proof. Let e be a closed covering of X. Since X is paracompact, we may choose a closed covering c such that c > e. Since (Ai) n-rc converges to A, we may choose a closed covering d and j e I such that drAi n> crAi for all i > jo Hence, from 2l1 we conclude that drA i >> erAi for all i > j Remarko For a regular topological space every open covering has an open star refinement if and only if the space is paracompact. We used this in the above proof in choosing c so that c > e. In this sense, a paracompact topological space provides a natural setting for applying 2. 2 to the theory of regular convergence. 2.8 Proposition. Let (Ai) be a net in C(X) n-rc converging to A. If e is a closed covering of X, there exists a closed covering b of X with b > e such that, for each closed covering c of X with c > b, there exists a closed covering d with d > c and j c I such that for all i > j 1. IHq(crAi Ai) Hq(Ai) for all q ~ n, 2 KHq(crA A i) C KH(cA I dr lAi) for all q ~ n + 1

-15Proof. Let e be a closed covering of Xo Choose closed coverings e0, e, o o, en, b of X and j' e I such that bA~Ai n-e nI Ai i erA i for all i>j' by 2. 7 Then with Ai=Ej for each j in 2.2, we have Hq(Ai)cIHq(brAi I Ai) for all q_ n and all i j'. Hence, IHq(cAi Ai) = H(Ai) for all q n and all i>j' since c>b. Choose closed coverings dl,.., dl, d of X and j" I such that d A in>> dn Ai.. >>dlAin>> cA for all i >j". Again by 2.2, we have KHq(crAi | Ai)cKHq(c_\Ai | drAi) for all q n+l and all i>j"o Choose j e I such that j >j' and j >j" to obtain the proposition 2.9 Proposition. Let (A.) be a net in C(X) n-rc converging to A, Let x e A and let E and D be closed nbds. of x with DCE~ If e is a closed covering of X, there exists a closed covering d of X and j e I such that dr\(Dr\Ai) n~ er(ErAi) for all i > j o Proof. Let e be a closed covering of X. Choose c a closed covering of X such that c >*eo Choose a closed nbd F of x such that DcF~ and FCE~o Form a new covering c' by replacing each element C of c that meets EO and X-E by the elements CrE and Cr(X-F~). Since (Ai) n-rc converges to A, we may choose a closed covering d with d >c' and j I such that dr~Ai r c'r'Ai for all i >j. Hence, dr_(DcAi)n> c'r,(ErAi)

-16for all i >j. But c' >*e so that we have that dnr(DrAi) n>> en(ErAi) for all i > j. 2 10 Proposition. Let (Ai) be a net in C(X) n-rc converging to Ao Let x e A and let E and D be closed nbds. of x with DcE~ If e is a closed covering of X, there exists a closed covering d of X with d > e and j e I such that for all i >j 1. IHq(ErNAi I DrAi)CIH4(d^(Dr-Ai) j Dr-Ai) for all q' n, 2. KH(er(ENAi) E^Ai)cKHqerEAi) dr(DrAi)) for all q n+l. Proof. Let x c A and let E, D, and e be as stated in the proposition. Choose closed sets E0, E, * * *, En,1 such that E CE~0 Ej CE~ for j = 1, 2, 2, * n-2, andDcEE Choose o 2 i i-I jC~~j~l n-ll closed coverings e e en-l d of X and j e I such that d-(DrAi) n>>en. (E_ lAi) >> * n>> eA e(EAi) n — ln- InA i - for all i >j by 2 9. Then from 2.2 we have that for all i > j IHq(EnAi I DrAi) c IHq(d_-DrCAi) I DrAi) for all q' n and KHq(erEreAi) j ErAi)C KHq(er(E^rAi) I en-^(En-_Ai)) for all q ~n+l. Since dr\(D^Ai) >en l (En lrAi), it follows that KHq(e(ErAi) | EaAi) cKHq(e (ErAi) | da(DAi)) for all i~ j and all q A n+1A

-172o11 Definition A closed covering e of X is in i i general position relative to a closed subset F of X if E ~r * onE qn F0 implies that (E )On * o.(E q)orF0O where E o, o E qare elements of e. 2.12 Proposition. Let e be a closed covering of a space X, and let F1, * *, Fs be closed subsets of X with FsC * CF1o Then there exists a closed covering d of X with d > e and such that d is in general position relative to Fr for r = 1, ~ ~, s. Proof. Let e be a closed covering of X. We may assume the elements of e are well ordered. Let El denote the first element in this ordering. For each0, ~, iq select a point x(i0, *, iq) i i in (E ~)~.n. (n(E q) (El)FO if possible. We, also, assume the point is chosen so that it is contained in the smallest possible Fro Since e is locally finite, there will be only a finite number of such points. Let G1 denote the union of the finite number of elements of e that meet El. Then E - G1 is an open subset of X contained in (El)~ Since X is normal, we may choose a closed set D1 such that (E -Gl)c(D )~, Dlc(E )~, and such that (Dl)~contains the above mentioned finite set of points. Then (D, E2, E3, ) is a closed covering of X, and DinE ~n ~ ~nE q^Fr0 for some r implies (D1)~n(El ~)~ On o (E q)o Fr )o A transfinite induction proof based on this construction shows there exists the desired covering. 2.13 Propositiono Let (Ai) be a net in C(X) converging to Ao Let F be a closed subset of X with F~YA/q0, and let

-18e be a closed covering of X in general position relative to FrAo Then there exists j e I such that erN(FrAi) and er~(FrA) may be identified for all i > j Proof. Let e and F be as in the proposition. Let u be the collection of open subsets of X consisting of F~, X-F, and elements of the form (Ei')Or ~ rN(Eiq)o where Eibr0 o Eir (FNA)/Eo Since (Ai) converges to A, there exists j e I such that Ai e N(u) for all i > j. But Ai e N(u) implies E1br0 ~ ElCr(FrA)#0 if and only i if E on~ ~ -E qn(F-Ai)10o Hence, we may make the assertion. Corollary. Let (Ai) be a net in C(X) converging to Ao Let e be a closed covering of X in general position relative to Ao Then there exists j e I such that erAi and erA may be identified for all i >j. 2014 Definition. If e is a collection of closed subsets of X and E is a closed subset of X, we let hq(er_(X-E)) denote the Cech relative group H (e, eriE). 2015 Lemma. Let E and D be closed subsets of X with DCE0. Let e be a closed covering of X. Then there exists a closed covering d of X such that Khq(e (X- E) cX-EKhq(er(X-E) | d(X-D)) for all q n where n is a fixed non-negative integer. Proof. It is clearly enough to show we can do this for any fixed q. Let e be a closed covering of X. Khq(e_(X-E) I X-E) is a submodule of a finitely generated L-module. Since L is a principal ideal domain, Kheq(_(X-E) X-E) is finitely generated, say by Zl' ~ ~,z os For each zr choose a closed covering cr of X with

-19cr> e such that zr Khq(ern(X-E) I c r(X-E))o Choose a closed covering d of X such that d>c for r= 1 ~ ~ ~, So Then Zr Khq(er(X-E) I dr(X-E)) for r = 1,~ ~ ~ s Since there is a natural homomorphism of hq(X-E) into hq(X-D), we have the assertion. Proposition. Let (Ai) be a net in C(X) n-rc converging to Ao Let x E A and let E and D be closed nbds. of x with DcE~. If e is a closed covering of X, there exists a closed covering d of X and j e I such that 1. Ihq(Ai-E I Ai-D)cIhq(d(Ai-D) I Ai-D) for all q n and all i >j. 2. Khq(e(Ai-E) I A-E)CKhq(er( Ai-E) I dr(A-D)) for all q _ n+l and all i > j Proof. Let x e A and let E, D, and e be as stated in the proposition. Choose F a closed nbd. of x with DCF~ and FCE~o Let e' = (E, X-F~)o Choose a closed covering b with the properties stated in 2. 8 relative to eo Choose c a closed covering of X with c >b, c >*e' and such that c is in general position relative to A, ErA, and DrAo Choose a closed covering d and j c I with the properties stated in 2.8 relative to c. Since this holds for any refinement of d, we may assume that d has the above general position properties and that d has the properties expressed in 2 10 if we substitute c for e in the proposition. By the lemma we may also assume that for a fixed i' with i' > j Kh(cr(Ai -E) Ai, - E)cKhcr(Ai, -E) I dn(A, -D))

-20for all q ~ n+l. If i > j and q _n+l, we have in the following diagram~ H (dr (Dr-Ai)) -h (dr (Ai-D)) Hqd Ai) P/' 1r' hq(c(Ai-E)) (qcrAi) hq(Ai- E) -3 -Hq(Ai) where the irs are natural homomorphisms and the rows are part of an exact cohomology sequence, that the kernel of 77 is contained in the kernel of ro We also have that the kernel of 77 is contained in the kernel of 7 for a fixed i' with i' > j and all q n+l. Let ljz= 0 for some z Fehq(c (Ai-E)). Then 7rz = 0 so that 7 T z = 0o Hence, $!z z 0 and by / exactness there exists w with /w= w 1T zo For the fixed i' mentioned above, //w = Oo But 9/ is independent of i by 2o13 for all i > j Hence 7z = 0 for all i >j and all q n+lo Since c.e, it follows Khq(er (Ai-E) j Ai-E)cKhq(er(Ai-E) | dr(Ai-D)) for all i > j and all q' n+lo Consider the following diagram: q-I. / q 1 /Hql(c(ErAi))-XA hq ( (Ai-E)) H Hq(crAi) Hq(cN (E rAi)) f (ErAi)- \ -hq(Ai-E) - Hq(Ai)} H( E Ai) il 1 1 w4 q q IT iq H-l (DA') i) h(Ai-D) -H (Ai) — -H (DAAi)

-21where the n's are natural homomorphisms and the rows are part of an exact cohomology sequence. We let Kir (1,) denote the kernel (image) of a homomorphism rt. Then if q ~ n and i j, we have I = 7 = Hq(Ai), K7CK, and I r I We show that I #cl~ I ^/' oLet x hq(Ai-E). If 3x = 0, choose y such that V3y = x. Choose z suchithat Z= // y. Then WI z= W4z/ z 7 = z y =T,y=' Xo If 3x O; choose u such that t-2 u =3Xo Then u =e3u u =<u3x O= 0 Hence5~u 0' Oso that 6/jU = 0. Choose v such that v /TUo Then 72-v = 77 u = U r7T3x = -/X Hence <4( 77/"v- 77;1 x) = 0. Now3x / 0 implies that a cocycle determining x is not the coboundary of an element of ErAi. But d > c implies A/ d >*e' so that 7/ v= 5 xo E. Eo Floyd [5] proved a homology version of the following theorem with L a field or compact group. Our proof is a cohomology version of his argument. 2o16 Theorem. Let (Ai) be a net in C(X) n-rc converging to Ao Then there exists j e I such that Hn(Ai) and Hn(A) are isomorphic for all i >j. Proof. Choose a closed covering b of X with the properties indicated in 2 80 Let c be a closed covering of X in general position relative to A and such that c > b Then by 2. 8 and 2 12 we may choose a closed covering d of X in general position relative to A with d > c and j' ~ I such that IHn(gcrAi I Ai) = Hn(Ai) and KHn(crAi | Ai) - KHn(cAi | drcAi ) if i >j Similarly, we may choose a closed covering f of X in general position relative to A with f >d and j" e I such that KHndAi A Ai) = KHnd(A |A IfAi)

-22if i > j"o Choose j e I with j j' and j >j" and such that the properties of 2.13 hold for c d, and f. Consider the following diagram: 17'/ Hn(Ai) Hn(frAi) = Hn(fA) -- Hn(A) S\\^,., ^, xHn @Ai) = H (dr-A) Hn(crA) = Hn (cA) where the fr's are natural homomorphisms and the indicated equalities hold by 2.13. Let z E Hn(A). We may assume that d was chosen so n that there exists w e H (dnA) with 4rw = z. Hence, there exists u e Hn(crAi) with 7u = 7Wo We may assume f.was-choseni s that 75 rr" u = /Wo Hence Ki2w =;2'w = =u = z. Hence IHn(cAA I A)=Hn(A). Let t7u = 0 for some u E Hn(crA). We may assume now thatf is chosen so that 27u =u 0, Then;T u = O so that T u = 0. Hence KHjn(cA I A) = KHn(crA I dNA) It follows that for i >j, Hn(Ai) and Hn(A) are each isomorphic with IHn(cA J drA)o The last half of this argument applies for dimension (n+l), also. Corollary. Let (Ai) be a net in C(X) n-rc converging to Ao If Hn+l(Ai) = 0 for all i, then Hn+l(A)= 0. 2.17 Theorem. Let (A.) be a net in C(X) n-rc converging to Ao Then A is clcn

-23Proofo Let x E A and let E be a closed nbd, of Xo Choose a closed nbdo F of x with FcE~o Let e = (E, X-F~) and choose a closed covering d of X and j' e I such that drAin> erAi for all i,j. Choose D e d such that x E D~o Then IHq(EnAi DnAi)=0 for all q.n and all i >j'o Choose a closed nbdo G of x with GCD~o We show that IHq(ErA | GnA) = 0 for all q n. Let c be a closed covering of X in general position relative to Er\A, DrA, and GrAo If z E Hq(EnAi) it may be represented by an element z(c) E Hq(cr(EnA)) for such a c. By 2 10 choose a closed covering d of X and j " e I such that KHq(cr(DnAi) 4 DnAi)cKHq(cr(DAi) dr (GnAi)) for all q n and all i j'"o We may assume d has the above general position properties of c. Choose j e I such that j > j' and j > j and such that the property expressed in 2.13 holds for the relevant nerves. Let 1i denote the homomorphism of Hq(crn(ErA)) into Hq(dr-(GrA)), 7. the homomorphism of Hq(cr-(EnAi)) into Hq(c r(DrAi)) and = the homomorphism of Hq(c r(DrAi.)) into Hq(d (GnAi)) where i > j. Observe that 7/ - 773 r/. Hence, zz(c), ^ 7z? 7z(c) = 0 for all q_ no Hence, z KHq(EnA I GnA) for all q n Corollary. Let U be an open subset of X with U compact Let (Ai) be a collection of compact subsets of U such that each point of U is in an element of some Ai and i > i' if and only if Ai, c Aio If for each x e U and each closed nbdo E of x, there exists a closed nbdo D of x with DcE and j e I such that IHq(ErAi I DnAi) = 0 for all i >j and all q n, then U- is clcn Remarko Eo Go Begle [1] claimed the homology version of the above theorem is not true for the Vietoris theory

-24with coefficients in a ring with unit element. 2o18 Definitiono For X locally compact andx E X, we write pr(x; X) = k, k a non-negative interger, if for each open nbd. U of x there exists open nbds. V and W of x with W C V, V- U, such that for any open nbd. W' of x with W' c W, hr(W' I V) = hr(W | V) and is a free L-module of rank k. 2919 Definition. Let U be an open subset of Xo For X locally compact, we say X has the n-dimensional Poincare duality property inside U if for each pair of open subsets P and Q with Q c P c U, IHn-q(p Q) is isomorphic with Ihq(Q I P) for all q n. For this definition we assume ordinary cohomology groups are used in dimension 0. 2.20 Theorem. Let X be locally compact and let (Ai) be a net in C(X) n-rc converging to Ao Suppose for each x e A there exists an open nbd. W of x and j e I such that Ai has the n-dimensional Poincare duality property inside Wr Ai for all i > j Then pr(x; A) = 0 for all x E A and all r< n, and pn(x; A)'1 for all x E Ao Proof. Let x E A and choose W and j' E I such that A. has the n-dimensional Poincare duality property inside W Ai for all i )j'o Let U be an open nbdo of x with U compact. Choose open nbdso P and Q of x with Q c P and P c (W r U) Choose an open nbd. V of x with V-c Q and j"e I such that IHq(QAr AJ V Ai) 0 for all q'n and all i>j" by the n-rc convergence. We show that Ihq(VrA I UN A) = 0 for all q <no Let e be a closed covering of X in general position relative to A,, A-U, A-P, A-Q, and A-V. If z e hq(V nA), it may be represented by an element z(e) e hq(e r (V A)) for such an eo By the proposition of 2 15, there exists a closed covering d of X and j"' c I such that

-25Khq(er-(PrAi) I PtrAi) CKhq(er (pAi) I d^(UNAi)) for all q _n+l and all i>j'"o We may assume that d has the above general position properties of e. Choose j E I such that j >j', j > j j >j"9 and such that the property expressed in 2 13 holds for the relevant nerves. Since IHn -(QCAi I Vr Ai) = 0 for all q <n and all iOj, the same is the case for Ihq(VcAi I QrAi) and hence for Ihq(VAAi I PrAi)o Hence, Ihq(e_1(VAi) | d_(UrAi)) = 0 for all q < n and all i > o Arguing as in 2 17, it follows that Ihq(V/rA j UrSA) = 0 for all q < n. Hence, pr (; A) = 0 for all x e A and all q< n. Since IH~ (QrAi I Vr-Ai) is isomorphic with L when ordinary cohomology groups are used, the above argument yields pn (x; A) 1 for all x e A.

CHAPTER III 300 We consider n-rc convergence of nets in this chapter where the elements of the net are generalized manifolds. 3.1 If X is locally compact, we write dim LX n if and only if h+l (U) = 0 for each open subset U of Xo This gives the cohomology dimension introduced by Cohen [3]. If dim LX -n, then hq(U) = hq(X-U) = 0 for all q _n+l and U an open subset of X. If dimLX is finite, then dim X' n if and only if pr(x; X) = 0 for all x X and all r n+l. A. Borel [2] contains a proof for thiso 3o2 Definition. A generalized n-manifold over L is a locally compact Hausdorff topological space M such that o1 dim L M is finite, 2. pr(x; M) = O for all x e M and all r f n, 3 pn(x; M) = 1 for all x e M We use the standard abbreviation n-gm for such a topological space. An orientable n-gm is an n-gm M such that for each component W of M and U a connected open subset of W with Ucompact, the natural homomorphism of hn(U) into hn(W) is an isomorphism. A locally orientable n-gm is an n-gm M such that each x e M has an open orientable n-gm for a neighborhood. Remark. If F is a proper closed subset of a connected locally orientable n-gm, then hn(F) = 0. If M is a connected n-gm, hn(F) = 0 for all proper closed subsets of M, and hn(M) is isomorphic with L, then M is an orientable n-gm. A. Borel [2] contains proofs for these statementso -26

-273 3 Proposition. Let X be locally compact with dim LX finite. Let (Ai) be a net in C(X) n-rc converging to A where dim L Ai - n for each i e Io Then (Ai) q-rc converges to A for any q and dim LA Ln. Proof. Let e be a closed covering of X. Choose a closed covering f of X such that f >e and each element of f is compact. Choose a closed covering d of X and j e I such that d.rA.n> f^A. for all i > j. Let 7denote the function from d into fo Then, by 3.1 Hq(/-DrA i) = hq(7rDrAi) = 0 for any q- n+l and any D E d. Hence, diAiq fA. for any q and all i > j Hence, drA.q> eA. for 1 - 1 1 any q and all i;j. Hence, (Ai) q-rc converges to A for any q. Let x A and let U be an open nbd. of x. Choose V and open nbd. of x with V C U. Using the proposition of 2.15, part 2., we may argue as in 2 20 to show that Ihq(VAA | UrA) - 0 for any fixed q with q n+l. Hence, pr(x; A) = 0 for all r n+l. Hence, dim L A-n by 3. 1. 3.4 Theoremo Let X be locally compact and let (Ai) n-rc converge to A where each Ai is a connected n-gm over L and dim LX is finite. Suppose for each x E A there exists an open nbd. W of x and j e I such that Ai has the n-dimensional Poincare duality property inside Wr-Ai for all i > j. Then A is a connected n-gm over Lo Proof. By o15, 2.20, 3.1, and 3.3, we need only show that pn(x; A) / 0 for all x E Ao Suppose pn(x; A) = 0 for some x e Ao Choose an open nbd W of x and j' e I such that Ai has the n-dimensional Poincare duality property inside WrAi. for all i >j'. Let U be an open nbd, of x 1~~~~~~~

-28with UcW and U- compact. Since p (x; A) = 0, there exists an open nbd P of x with Pc U such that Ihn(PrA I UrA) = O. By the n-rc convergence choose an open nbdo V of x with V c P and j" e I such that H~(UcrAi I Vr\Ai) is isomorphic with L for all i > j" where ordinary cohomology groups are usedo Since U and V are contained in W, Ihn(VrAi I UreAi) is isomorphic with L for all i > j". By part 1. of the proposition in 2.15, we may choose a closed covering e of X in general position relative to A-U, A-P, and A-V so that Ihn(VrAi I PAi) c Ihn(er,(PeAi) I Pr-Ai) for large i. By the lemma in 2015, we may choose a closed covering d of X with the above general position properties so that Khn(er(UrA) | UrA)c Khn(e_(UrtA) I dcr(UrA)). Hence, KhnlerN(P^A) U^A) Khn(e(PrA) I dr(U^A)). But Ihn(er(PcA) I UrA)= 0 by the choice of P. Hence, for large i Ihn(erN(PrAi.) | ds(UrAi)) = 0. This together with the above assertion concerning Ihn(V^Ai I Pr-Ai) contradicts the fact that Ihn(VrAi UrAi) f 0 for large io 3.5 Theorem. Let X be locally compact with dim LX finite. Let (Ai) be a net in C(X) (n-l)-rc converging to A where each Ai is a connected n-gm over a field L and A is non-degenerate. Suppose for each x E A there exists an open nbd. W of x such that WrAi is an orientable n-gm over L for all i e Io Then A is a connected n-gm over Lo Proof. We first show that (Ai) n-rc converges to A. Let e be a closed covering of Xo Choose a closed covering f with each of its elements compact, f )e, and such that for each F e f with F~rcA = 0 there exists a W as in the hypothesis with F c WrE~ and

-29(WrA)-F; Ofor some E e e. Choose a closed covering d of X with d >f andj' E I such that d-Ain- 1 frAi for all i j'. Since (Ai) converges to A in C(X), there exists j e I with j > j' such that (W^Ai)-D # 0 for some W where D~OA A 0 provided i; j. Hence, Hn(D-Ai) = hn(DrAi) = 0 for all i >j and all D E d. Hence, drAin> erNAi for all i > j Since an orientable n-gm over a field has the n-dimensional Poincare duality property, it follows from 3.4 that A is a connected n-gm over Lo 3. 6 Theorem. Let X be compact with dim LX finite Let (Ai) be a net in C(X) (n-l)-rc converging to A where each Ai is a connected compact orientable n-gm over a field L and A is non-degenerate. Then A is a connected compact orientable n-gm over Lo Proof. By 3.5, A is a connected compact n-gm over L.o We show that A is orientable. Let D be a closed nbdo of some x e A with A-D f 0. We show that hn(DNA) = Hn(DrA) = 0. Since dim LA = n, this will follow provided IHn(A I DrA) = 0. Choose E a closed nbd. of x with D~C E and A-E f 0. An element of Hn(A) would be determined by a closed covering e of X in general position relative to ErnA and DnAo Choose a closed covering d of X.with these general position properties such that KHn(er(ErAI) I EAi) C KHn(er(ErAi) | dN(DnAi)) for large i be 2.10. Since Ai is orientable, hn(EnAi) = Hn(ErAi) = 0 i 1 for large i. Hence, for large i KHn(eN(ENAi) I dr(DnA)) = Hn(e(EoAi)). Hence, for large i IHn(e_~Ai I dr(DrAi)) = 0o Hence,

-30IHn(er-A dr(DrA)) = 0o Hence, IHn(A DNA) = 0. By 2.16, hn(A) = Hn(A) is isomorphic with Hn(Ai) = hn(Ai) for large i. Hence, hn(A) is isomorphic with L. Hence, A is orientable by the remark in 3.2o Remark. This theorem generalizes the result of E. G. Begle [1] mentioned in the introduction. Notice that the requirement that A be n-dimensional is not necessary. P. A. White [10] gives a proof for this theorem with the assumption that dim LA = no However, as mentioned in the introduction, his definition of regular convergence is not proven to be equivalent with the standard definition. Example. Let X be the plane and let A. denote the set points (x, y) such that xZ + y2 (1/i) where i ranges over the positive integers. Then (Ai) 0-rc converges to the point (0, 0). Hence, the above non-degeneracy assumption is necessary. 3o7 Definition. By an n-gm with boundary B we mean a locally compact Hausdorff topological space M and a closed subset B such that 1o B is an (n-l)-gm, 2. M-B is an n-gm, 3o pr(x; M) = 0, for all x E B and all r. For convenience, we say that M is a 1o Oo n-gm with lo o, boundary B, when M is an n-gm with boundary B and M-B and B are locally orientable o This definition is given by Fo Ao Raymond [8] and the following two propositions proven by him.

-313.8 Proposition. Let Mi be a.l oo (orientable) n-gm with 1. oo (orientable) boundary Bi for i = 1, 2. If M - MIpM2 and MlrM2 = B1 = B2, then M is a locally orientable (orientable) n-gm. 3o 9 Proposition. Let M be a connected locally orientable (orientable) n-gm, and let M' be a connected locally orientable (orientable) (n-l)-gm imbedded as a closed subset of M. If M-M' is separated, then M-M' is the union of exactly two disjoint connected open sets each of whose frontiers is M', and onto each of which M' fits as a 1o Oo (orientable) n-gm with 1o Oo (orientable) boundary. 3o10 Proposition. Let (Ai) and (Bi) be nets in C(X) n-rc converging to A and B respectively. If (AirBi) (n-l)-rc converges to A\ B, then (A.lBi) n-rc converges to A~Bo Proofo Let e be a closed covering of X. Choose a closed covering f of X and jl I such tnat flAAin> erA for all i >jlo Choose a closed covering f2 of X and j2 ~ I such that rf-Bir^e^Bi for all i j2o Choose a closed covering f such that f f and f >f2 Choose a closed covering d of X and j E I with J > j, and j >2 such that dr(A.rBi ) n- fo(Ai.NBi) for all i > j Let rdenote the map of d into f and r the map of linto e For any D E d and i > j consider the following diagram: Hq(EN(Ai Bi)) -Hq (EAi) @ Hq(Er-Bi) H1 / A q 1 1 1 1 1 I Hq 1 1 (A Bj)) -^Hq D(A BiH) (F —, BDB F% AB, B g (i-N-)6DH FrB.

-32where f = irD, E = i F, the rows are a portion of an exact Mayer-Vietoris cohomology sequence, and the vertical homomorphisms are natural. If q n and i >j, the homomorphisms, and V have trivial images. Using this and the exactness of the rows, it follows that the image of 2 eO/ is trivial for all q ~n and all i >j. Hence, the map zrirof d into e gives that drnAiBi) neN(Ai.Bi) for all i>jo Hence, (Ai.Bi) n-rc converges to ABo 3.11 Theorem. Let X be compact with dim LX finite and let (Ai) be a net in C(X) (n-l)-rc converging to a non-degenerate set A where each Ai is a connected compact orientable n-gm with 1. o. boundary Bi over a field L. If (Bi) (n-2)-rc converges to a non-degenerate set B, A is a connected compact orientable n-gm with connected orientable boundary B over L. Proof. For each i let Ci denote the double of Ai formed by attaching two distinct copies of Ai along Bi and let C denote the double of A. F. Ao Raymond [8] shows that in the above situation each Bi is actually connected and orientable. Hence, by 3.8 Ci is a connected compact orientable n-gm. By 3.10, (Ci) n-rc converges to C. Hence, C is a connected compact orientable n-gm. Similarly, B is a connected compact orientable (n-l)-gm. Since B separates C, A is a connected compact orientable n-gm with connected orientable boundary B by 3. 9 Remark. P. A. White [11], using the paper referred to in 3 6, proved this result with the additional assumption that dim L A= no Remark. It is not known whether a n-gm is always

-33locally orientable. Should this be the case, 3. 5 together with the above argument would give a similar result for the non-compact caseo Since a separable metric n-gm is a classical manifold for n 2, it is locally orientableo Hence, we may state the following result. 3.12 Theorem. Let X be locally compact and metrizable with dim LX finite. Let (Ai) be a net in C(X) 1-rc converging to a non-degenerate set A where each Ai is a classical connected 2-manifold with connected boundary Bi. Suppose for each x e A there exists an open nbd. W of x such that WrNAi is a classical orientable 2- manifold with boundary WrBi for all i e I. If (Bi) Q-rc converges to a non-degenerate set B, then A is a connected 2-manifold with connected boundary B provided A is separableo

BIBLIOGRAPHY 1. Eo Go Begle, Regular convergence, Duke Math. Jouro, Vol. 11 (1944), pp. 441-450. 2. A. Borel, Seminar on Transformation Groups, Princeton, 1960. 3. H. Cohen, A cohomological definition of dimension for locally compact Hausdorff spaces, Duke Math. Jour., Vol. 21 (1954), pp, 209-224 4o E. Dyer, Regular mappings and dimension, Ann. of Math., Volo 67 (1958), ppo 119-149. 5. Eo Eo Floyd, Closed coverings in Cech homology theory, Trans. Amero Math. Soc., Volo 14 (1957), pp. 319-339. 6. J. L. Kelley, General Topology, Van Nostrand, 1955. 7. Eo Michael, Topologies on spaces of subsets, Trans. Amero Math. SoCo, Volo 71 (1951), ppo 151-182. 8. Fo A. Raymond, Separation and union theorems for generalized manifolds with boundary, Michigan Math. Jouro, Vol. 7 (1960), ppo 7-21. 9o L. Vietoris, Bereiche zweiter ordnung, Monatshefte fur Mathematik und Physik, Volo 33, (1923), pp. 49-62. V 10. P. Ao White, Regular convergence in terms of Cech cycles, Ann. of Matho Volo 55 (1952), ppo 420-432..llo --—, Regular convergence of manifolds with boundary, Proc. Amero Matho SoCo, Volo 4 (1953), ppo 482-485. 12o —, Regular convergence, Bullo Amero Math. Soc., Vol. 60 (1954), ppo 431-443. 13. G. To Whyburn, On sequences and limiting sets, Fund. Math., Volo 25 (1935), ppo 408-426. 14. Ro Lo Wilder, Topology of Manifolds, Amero Matho Soc., Collogquium Publication, Volo 32, New York, 1949. 15. Go So Young, Jro, On 1-regular convergence of sequences of 2-manifolds, Amero Jouro of Math,, Volo 71 (1949), ppo 339-348. 16. Co Zarankiewicz, Sur les points de division dans les ensembles connexes, Fundo Math., Volo 9 (1927), ppo 124-171. -34

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