Advances in Pure Mathematics, 2012, 2, 296300 http://dx.doi.org/10.4236/apm.2012.24039 Published Online July 2012 (http://www.SciRP.org/journal/apm) A ℘x and Open C D C Filters Process of Compactifications and Any Hausdorff Compactification Hueytzen J. Wu1, WanHong Wu2 1Department of Mathematics, Texas A & M UniversityKingsville, Kingsville, USA 2University of Texas at San Antonio, One UTSA Circle, San Antonio, USA Email: hueytzen.wu@tamuk.edu, dd1273@yahoo.com Received February 23, 2012; revised March 15, 2012; accepted March 22, 2012 ABSTRACT By means of a characterization of compact spaces in terms of open filters induced by a D C*(Y), a ℘x and open filters process of compactifications of an arbitrary topological space Y is obtained in Sec. 3 by embedding Y as a dense subspace of (, ℑℬ) or (Y, ℑℬ), where = YE YS, = YE YT, YE = {NxNx is a ℘xfilter, x Y}, YS = {ℰℰ is an open filter that does not converge in Y}, YT = {ÅÅ is a basic open D C YW T Y W S Y D C W T W S D C filter that does not converge in Y}, ℑℬ is the topology induced by the base ℬ = {U*U is open in Y, U } and U* = {ℱ (or Y)U ℱ}. Furthermore, an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X can be obtained from a D C*(X) by the similar process in Sec. 3. W S YW T D CD C Filter; Open Keywords: Net; Open Filter; Open Filter Base; Basic Open D C Filter; ℘Filter; ℘xFilter; Tychonoff Space; Normal T1Space; Compact Space; Compactifications; StoneČech Compactification; Wallman Compactification 1. Introduction Throughout this paper, [T]< denotes the collection of all finite subsets of the set T. For the other notations and ter minologies in General Topology which are not explicitly defined in this paper, the readers will be referred to the Ref. [1]. For an arbitrary topological space Y, let C*(Y) be the set of bounded realvalued continuous functions on Y, D C*(Y). It is shown in Sec. 2 that there exists a unique rf Cl(f(Y)) for each f in D such that for any H [D]< , > 0, fHf−1((rf − , rf + )) . Let Vr ={fHf−1((rf − , rf + ))fHf−1((rf − , rf + )) for any H [D]< , > 0}. Vr is called an open filter base. An open filter ℰr on Y containing an open Dfilter base Vr is called an open filter. An open filter År on Y generated by an open Dfilter base Vr is called a basic open D D C C D C CC filter. By a characterization of compact spaces in Sec. 2 and the ℘x and open Dfilters process of compactifications in Sec. 3, Y can be embedded as a dense subspace of (S Y, ℑℬ) or (T Y, ℑℬ), where S = YE YS, T = YE YT, YE = {NxNx is a ℘xfilter, x Y}, YS = {ℰℰ is an open Dfilter that does not converge in Y}, YT = {ÅÅ is a basic open filter that does not converge in Y}, ℑℬ is the topology induced by the base ℬ = {U*U , U is open in Y} and U* = { ℱ S (or T Y)U ℱ}. Furthmore an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X can be obtained from a D C*(X) by the similar process in Sec. 3. C W W YW Y C D C W YW C W 2. Open D Filters and a Characterization of Compact Spaces Let Y, C*(Y) and D be the sets that are defined in Sec. 1. Theorem 2.1 Let ℱ be a filter on a topological space Y. For each f D, there exists a rf Cl(f(Y)) such that f−1((rf − , rf + )) F for any F ℱ and any > 0. (See Thm 2.1 in the Ref. [2, p. 1164].) Proof. If the conclusion is not true, then there is an f D such that for each rt Cl(f(Y)), there exist an Ft ℱ and an t > 0 such that Ft f−1((rt − t, rt + t)) = . Since Cl(f(Y)) is compact and Cl(f(Y)) {(rt − t, rt + t)rt Cl(f(Y))}, there exist r1, ···, rn in Cl(f(Y)) such that Y = f−1(Cl(f(Y))) = {f−1((ri − i, ri + i))i = 1, ···, n}. Let Fo = {Fi i = 1, ···, n}, then Fo ℱ and Fo = Fo Y {[ Fi f−1((ri − i, ri + i))]i = 1, ···, n} = , contradicting that ℱ. Corollary 2.2 Let Q be an open ultrafilter on Y. For C opyright © 2012 SciRes. APM
H. J. WU, W.H. WU 297 each f D, there exists a unique rf Cl(f(Y)) such that (1) for any H [D] , any > 0, fHf−1((rf − , rf + )) Q and (2) for any H [D] , any > 0, fHf−1((rf − , rf + )) . (See Cor. 2.2 in the Ref. [2, p. 1164].) Therefore, for a given open ultrafilter Q, Q contains a unique open filter base Vr = {fHf−1((rf − , rf + )) fHf−1((rf − , rf + )) for any H in [D]< , > 0}. Vr is called an open Dfilter base. An open filter ℰr on Y containing an open Dfilter base Vr is called an open filter. An open filter År on Y generated by an open Dfilter base Vr is called a basic open Dfilter. For each f D, if rf = f(x) for an x in Y, then Vr and År are called the open Dfilter base and the basic open filter at x, denoted by Vx and Åx, respectively. C C C C C C C C C C C D C C D C C D D Definition 2.3 Let L be a family of continuous functions on Y. A net {xi} in Y is called a Lnet, iff {f(xi)} converges for each f L. Theorem 2.4 Let L be a set of continuous functions on Y. Then Y is compact iff (1) f(Y) is contained in a com pact set Cf for each f in L, and (2) every Lnet has a cluster point in Y. Proof. Let {xi} be an ultranet in Y. For each f in L, {f(xi)} is an ultranet in Cf, hence converges in Cf; i.e., {xi} is a Lnet. (2) implies that {xi} has a cluster point x in Y. Since {xi} is an ultranet, {xi} converges to x. Thus, Y is compact. The converse is obvious. Corollary 2.5 Let D C*(Y). If every Dnet converges in Y, then Y is compact. Definition 2.6 If ℱ is a filter on Y, let ℱ = {(x, F)x F ℱ}. Then ℱ is directed by the relation (x1, F1) (x2, F2) iff F2 F1, so the map P: ℱ Y defined by P(x, F) = x is a net in Y. It is called the net based on ℱ. (See Def.12.16 in the Ref. [1, p. 81].) Corollary 2.7 If ℱ is a filter on Y, {P(x, F)} is the net based on ℱ, then ℱ = {S YP(x, F)} is eventually in S}. (See L2) in the Ref. [3, p. 83].) Lemma 2.8 Let D C*(Y). 1) For each open Dfilter ℰ, let Vr, as the Vr defined in Section 1, be an open Dfilter base such that Vr ℰ. Then the net {xF} based on ℰ is a Dnet such that lim{f(xF)} = rf for each f D. 2) For each Dnet {xi} in Y, {xi} induces a unique open filer base V{xi} on Y. D Proof. 1) By Cor. 2.7, {xF} is eventually in f−1((rf − , rf + )) Vr ℰ for each f D and any > 0. Thus lim{f(xF)} = rf for each f D; i.e., {xF} is a Dnet. 2) Let {xi} be a D net. For each f D, let tf = lim{f(xi)}. Then fHf−1((tf − , tf + )) for any H [D]< , any > 0. Let V{xi} = {fHf−1((tf − , tf + ))fHf−1((tf − , tf + )) for any H [D]< , > 0}, then V{xi} is an open Dfilter base on Y. Since tf is unique for each f D, thus V{xi} is uniquely induced by {xi}. Theorem 2.9 Let D C*(Y). Then, 1) and 2) in the following are equivalent: 1) Every Dnet converges in Y. 2) Every open filter ℰ converges in Y. Proof. 1) 2) is obvious by Lemma 2.8 1) above and Thm. 12.17 (a) in the Ref. [1, p. 81]. For 2) 1): Let {xi} be a Dnet in Y, let ℱ = {O is open and {xi} is eventu ally in O}. Clearly, ℱ is an open filter. For each f in D, let tf = lim{f(xi)}, then {xi}is eventually in f−1((tf − , tf + )) for any > 0; i.e., for each f in D, any > 0, f−1((tf − , tf + )) ℱ, so ℱ is an open Dfilter. 2) implies that ℱ converges to a point x. Thus, for any open nhood Ux of x, Ux ℱ; i.e., {xi} is eventually in Ux. So {xi} converges to x. Corollary 2.10 If every open filter ℰ on Y con verges in Y, then Y is compact. 3. An Open D Filter Process of Compactification For each x Y, let Nx = {{x}} {OO is open, x O}. Nx is a ℘filter (See 12E. in the Ref. [1, p. 83] for its definition and convergence.) with = Nx. For each x Y, Nx is called a xfilter. Let YE = {NxNx is a ℘xfilter, x Y}, YS = {ℰℰ is an open D C filter that does not converge in Y}, YT = {ÅÅ is a basic open Dfilter that does not converge in Y}, Y = YE YS and = YE YT. C W W Y W YW C W YW W W Y W S YW Y W YW Y W YW W W S T Lemma 3.11 For each ℱ S (or T Y), there is a unique rf Cl(f(Y)) for each f D such that f−1(rf − , rf + ) Vr ℱ for all > 0. Proof. If ℱ = Nx for an x Y, then for each f D, f−1((rf − , rf + )) Vx Nx for all > 0, where rf = f(x). If ℱ = ℰ (or Å), then there is an open Dfilter base Vr, as the Vr defined in Sec. 1, such that for each f D, f−1((rf − , rf + )) Vr ℰ (or Å) for all > 0. The uniqueness of rf for each f D follows from Cor. 2.2. Definition 3.12 For each open set U in Y, define U* = {ℱ (or Y)U ℱ}. S T Lemma 3.13 1) For any open set U in Y, U U* ; 2) U = Y U* = S Y (or T); and 3) for any ℱ in (or ), any open set U in Y, ℱ U* U ℱ. T Proof. 1) If U , pick an x U, then U Nx Nx U*; i.e., U* . If U* , pick a ℱ U*, then U ℱ U . 2) and 3) are obvious from Def. 3.12. Lemma 3.14 For any two nonempty open sets S and T in Y, 1) S T iff S* T*, and 2) (S T)* = S* T*, if S T . Proof. 1): (): ℱ S* S ℱ T ℱ ℱ T*. (): S ⊈ T there is a y (S – T) Ny (S* − T*) S* ⊈ T*. 2): By 1) above, (S T)* S* T*. If ℱ S* T*, then S ℱ, T ℱ and S T ℱ; i.e., ℱ (S T)*. Proposition 3.15 ℬ = {U*U is an open set in Y} is a base for (or ). S T Proof. For (a) in Thm. 5.3 in the Ref. [1, p. 38]: For each ℱ S (or T Y), pick a O ℱ. Then O , ℱ O* and O* ℬ. Thus S Y (or T Y) = {U*U* ℬ}. For (b): If ℱ S* T* for S*, T* ℬ, then S, T ℱ, Copyright © 2012 SciRes. APM
H. J. WU, W.H. WU 298 S T ℱ, (S T)* ℬ and ℱ (S T)* S* T* ℬ. W W W W W W W S Y W S YW Y W W S W T Y W YW W YW WW W W T W W* V C W Y W C Equip S Y (or T Y) with the topology induced by ℬ. For each f in D, define f*:S Y (or T Y) by f*(ℱ) = rf, if f−1((rf − , rf + )) Vr ℱ for all > 0. By Lemma 3.11, for each f D, f* is welldefined and f*(S Y) (or f*(T Y) Cl(f(Y)), thus f* is a bounded realvalued function on (or Y) such that f*(Nx) = f(x) for all x Y. W T Proposition 3.16 For each f in D, let t Cl(f(Y)). For any , with 0 < < , 1) [f−1((t − , t + ))]* f*−1((t − , t + )), 2) f*−1((t − , t + )) [f−1((t, t+))]*. Proof. 1): If ℱ [f−1((t − , t + ))]*, then f−1((t − , t + )) ℱ. If f*(ℱ) = e, then f−1((e − , e + )) ℱ for all > 0. Since f−1((t − , t + ) (e − , e + )) = f−1((t − , t + )) f−1((e − , e + )) ℱ for all > 0, so (t − , t + ) (e − , e + ) for all > 0. Thus f*(ℱ) = e [t − , t+] (t − , t + ); i.e., ℱ f*−1((t − , t + )). 2): If ℱ f*−1((t − , t + )), then f*(ℱ) = s (t − , t + ) and f−1((s − , s + )) ℱ for all > 0. Pick > 0 such that (s − , s + ) (t − , t + ). Then S = f−1((s − , s + )) f−1((t − , t + )) and S ℱ. Thus f−1((t − , t + )) ℱ; i.e., ℱ [f−1((t − , t + ))]*. Proposition 3.17 For each f D, f* is a bounded real valued continuous function on (or ). T Proof. For any ℱ S (or T Y), let f*(ℱ) = t. We show that for any > 0, there is a U* ℬ such that ℱ U* f*−1((t − , t + )). Let U = f−1((t − /2, t + /2)). Since f−1((t − , t + )) ℱ for all > 0. Thus, U = f−1((t − /2, t + /2)) ℱ; i.e., ℱ U*. By Prop. 3.16 1), ℱ U* f*−1((t − , t + )). Thus f* is continuous on Y (or ). W Y Lemma 3.18 Let k: Y S (or T) be defined by k(x) = Nx. Then, 1) k is welldefined, oneone and k−1(U*) = U for all nonempty open set U in Y and all U* ℬ; i.e., k is continuous; 2) f* o k = f for all f D; 3) k(Y) is dense in (or Y). Y S T Proof. 1) For any x, y in Y, x = y Nx = Ny, thus x y Nx Ny, so k is welldefined and oneone. For any U* ℬ, by Def. 3.12 and Lemma 3.13 1), U* , U is open, U . So (a): x k−1(U*) (b): Nx = k(x) U*. By Lemma 3.13 3), (b) (c): U Nx. By the setting of Nx, (c) (d): x U. Thus k−1(U*) = U for all U* ℬ, U and U is open in Y; i.e., k is continuous. 2) is obvious from (f* o k)(x) = f*(Nx) = f(x) for all x in Y and all f in D. 3) For any U* ℬ, pick a ℱ U*, then U ℱ and U . Pick an x U, by 1) above, x U k(x) U*; i.e., k(x) U* k(Y) . Hence k(Y) is dense in (or Y). S T Let D* = {f*f D}. Then, D* C*(S Y) (or C*(Y)). For each open C*D*filter ℰt* on S Y (or T Y), let t = {f*H*f*−1((tf* − , tf* + ))f*H*f*−1((tf* − , tf* + )) for any H* [D*]< , > 0} be the open Dfilter base on S (or ) such that ℰ* t. Since f* o k = f, k is oneone and k(Y) is dense in S Y (or ), so k(fHf−1 ((tf* − , tf* + ))) = [f*H*f*−1((tf* − , tf* + ))] k(Y) for any H* [D*]< , H = {f Df* H*} and any > 0. Thus Vt = {fHf−1((tf* − , tf* + ))fHf−1((tf* − , tf* + )) for any H [D]< , > 0} is a welldefined open D filter base on Y. Let ℒS = {U Y  U is open, U and U* ℰ* t} and ℒT = Åt, the basic open D C filter generated by Vt. Since ℰ* t is a filter, clearly, by Lemma 3.14, ℒS is an open filter on Y. Y W T Y* t V W T Y Lemma 3.19 ℒS is an open filter on Y. D Proof. For any H [D]< , > 0, let H* ={f*f H}, O = fHf−1((tf* − , tf* + )) and P = f*H*f*−1((tf* − , tf* + )). Then P t ℰ* t. By Lemmas 3.13, 14 and Prop. 3.16 2), P O*, O* ℰ* t, O and O ℒS. This implies that Vt ℒS. C * V W *W W Y * W YW W S W W Y W Y W W W T C Theorem 3.20 (Y, k) is a compactification of Y. S Proof. Case 1: If ℒS converges to a point p in Y. Let U be any open set in Y such that k(p) U* ℬ. By Lemma 3.18 1), p U = k−1(U*), thus U ℒS; i.e., U* ℰ* t. This implies that ℰt converges to k(p) in S Y. Case 2: If ℒS does not converge in Y, then ℒS S. For any U* in ℬ such that ℒS U*, U ℒS and therefore U* ℰt. This shows that ℰ* t converges to ℒS in S. By Cor. 2.10, S Y is compact and by Lemma 3.18 3), (Y, k) is a compac tification of Y. Lemma 3.21 For each open set U ℒT = Åt, U* ℰ* t. Proof. If U Åt, then there exist a H [D]< , an > 0 such that E = fHf−1((tf* − , tf* + )) Vt and E U. Lemma 3.14 and Prop. 3.16 2) imply that F = f*H*f*−1 ((tf* − , tf* + )) E* U* and F ℰ* t. Thus, U* ℰ* t. Theorem 3.22 (Y, k) is a compactification of Y. T Proof. Case 1: If ℒT = Åt converges to a point p in Y, let U be any open set in Y such that k(p) U*, Lemma 3.18 1) implies that p U, thus U ℒT = Åt. So by Lemma 3.21, U* ℰ* t. This implies that ℰ* t converges to k(p) in T. Case 2: If ℒT = Åt does not converge in Y, then ℒT = Åt T. For any U* ℬ such that Åt U*, U Åt and by Lemmas 3.21, U* ℰ* t. Thus ℰ* t converges to ℒT = Åt in T Y. Cor.2.10 implies that T Y is compact and by Lemma 3.18 3), (Y, k) is a compactification of Y. 4. An Arbitrary Hausdorff Compactification of a Tychonoff Space For an arbitrary Hausdorff compactification (Z, h) of a Tychonoff space X, let D = {ff = ˚f o h, ˚f ˚D = C(Z)}. Then D C*(X), D separates points of X and the topology on X is the weak topology induced by D. For each x X, let Vx, as the Vx defined in Section 2, be the open D filter base at x induced by D. Obviously, we can easily get Lemma 4.21 as follows: Lemma 4.21 GD = {Vxx X} is a base for the topology on X and for each x X, Vx is an open nhood base at x. Copyright © 2012 SciRes. APM
H. J. WU, W.H. WU 299 Let XW = {ÅÅ is a basic open Dfilter on X}. For each År XW, let Vr, as the Vr defined in Sec. 1, be the open Dfilter base that generates År. If År converges to an x X, then for each f D, x Cl(f−1((rf − /2, rf + /2)) f−1([rf − /2, rf + /2]) f−1((rf − , rf + )) for all > 0; i.e., rf = f(x) for all f D, so Vr = Vx and År = Åx. Thus XW = XE XF and XE XF = , where XE = {Åxx X} and XF = {ÅÅ is a basic open Dfilter that does not converge in X}. Similar to what we have done in Section 3, we can get the similar definitions and results for XW in the following: C C C C C D C * t V C C 4.221. For each open set U in X, define U* = {Å XWU Å}. 4.222. 1) for any open set U in X, U U* ; 2) U = X U* = XW; and (c) for any Å in XW, any open set U , Å U* U Å. 4.223. For any two nonempty open sets S and T in X, 1) S T iff S* T*, and 2) (S T)* = S* T*, if S T . 4.224. ℬ ={U*U , U is an open set in X} is a base for a topology on X. 4.225. For each f D, f*: XW is defined by f*(År) = rf, if f−1((rf − , rf + )) Vr År for all > 0. Then f*(Åx) = f(x) for all x X. 4.226. For each f in D, let t Cl(f(X)). For any , with 0 < < , 1) [f−1((t − , t + ))]* f*−1((t − , t + )), 2) f*−1((t − , t + )) [f−1((t − , t + ))]*. 4.227. For each f in D, f* is a bounded realvalued continuous function on XW. 4.228. Define k: X XW by k(x) = Åx, then 1) k is welldefined, oneone, and U = k−1(U*) for all open set U in X and all U* ℬ; i.e., k is continuous, 2) f* o k = f for all f in D and 3) k(X) is dense in XW. 4.229. Let D* = {f*f D}. Then D* C*(XW). Lemma 4.23 D* separates points of XW. Proof. For Ås, Åt XW, let Vs = {fHf−1((sf − , sf + ))fHf−1((sf − , sf + )) for any H [D]< , > 0} be the open Dfilter base that generates Ås and similarly for Vt. Since Ås = Åt, Vs = Vt and that sf = tf for all f in D are equivalent, thus Ås Åt, Vs Vt and that there is a g in D such that sg tg are equivalent. So, if Ås Åt, then g*(Ås) = sg tg = g*(Åt) for some g* D*. Lemma 4.24 The topology on XW is the weak topology induced by D*. Proof. For each År XW, let Vr, as the Vr defined in Sec. 1, be the open Dfilter base that generates År and let U* ℬ such that År U*, then U År. So there exist a H [D] , an > 0 such that fHf−1((rf − , rf + )) U, where fHf−1((rf − , rf + )) Vr År for all > 0. By 4.222 (c), 4.223 and 4.226 2), År [fHf−1((rf − /2, rf + /2))]* f*H*f*−1((rf − , rf + )) [fHf−1((rf − , rf + ))]* U*; i.e., År f*H*f*−1((rf − , rf + )) U*. For any open filter ℰ on XW, let = {f*H* f*−1((tf* − , tf* + ))f*H*f*−1((tf* − , tf* + )) for any H* [D*]< , > 0} be the open Dfilter base that is contained in ℰ* t. Since f* o k = f for all f D, k is one one and k(X) is dense in XW, so k(fHf−1((tf* − , tf* + ))) = f*H*f*−1((tf* − , tf* + )) k(X) for any H* [D*]< , H = {f Df* H*}) and any > 0. Thus Vt = {fHf−1((tf* − , tf* + ))fHf−1((tf* − , tf* + )) for any H [D]< , > 0} is a welldefined open D * t filter base on X. Let Åt be the basic open Cfilter on X generated by Vt. D * V C Lemma 4.25 For any open set U Åt, U* ℰ* t. Proof. For any U Åt, there exist a H [D] , an > 0 such that fHf−1((tf* − , tf* + )) = S Vt and S U. By 4.223 and 4.226, T = f*H*f*−1((tf* − , tf* + )) S* U* and T . Thus U* ℰ* t. t Theorem 4.26 (XW, k) is a Hausdorff compactification of X. Proof. We show that the open Dfilter ℰ* t converges to Åt in XW. For any open set U in X such that Åt U*, by 4.222 (c), U Åt, by Lemma 4.25, U* ℰ* t. This implies that ℰ* t converges to Åt in XW. By Cor. 2.10, XW is compact. Thus, by 4.228 3) and Lemma 4.23, (XW, k) is a Hausdorff compactification of X. 5. The Homeomorphism between (XW, k) and (Z, h) For each basic open D C filter År XW, let Vr, as the Vr defined in Sec. 1, be the open Dfilter base that gener ates År. Since h−1: h(X) X is oneone, f = ˚f o h and h(X) is dense in Z, so h−1(˚f˚H˚f−1((rf − , rf + ))) = fHf−1 ((rf − , rf + )) for any ˚H [˚D]< , H = {f˚f ˚H} and any > 0. Thus, ˚Vr = {°f°H˚f−1((rf − , rf + )) ˚f˚H˚f−1((rf − , rf + )) for any ˚H [˚D] , > 0} is a welldefined open D C C filter base on Z. Let ˚År be the basic open D C filter on Z generated by ˚Vr. Since Z is compact, ˚År clusters at a zr Z. For each ˚f ˚D, zr Cl(˚f−1((rf − /2, rf + /2))) ˚f−1([rf − /2, rf + /2]) ˚f−1((rf − , rf + )) ˚Vr for all > 0; i.e., ˚f(zr) = rf for all ˚f ˚D. So ˚Vr = ˚Vzr and ˚År = ˚Åzr. The zr is called the w point in Z induced by År such that ˚f(zr) = rf = f*(År) for all ˚f ˚D and f* D*. ˚Vzr and ˚Åzr are called the open D C filter base and the basic open Dfilter at zr in Z induced by Vr or År, If zs zr in Z, there is a ˚f ˚D such that ˚f(zs) ˚f(zr) = rf = f*(År), so zr is the unique wpoint in Z induced by År. If Åt År, let zt be the wpoint in Z induced by Åt. By Lemma 4.23, there is a g* D* such that ˚g(zt) = g*(Åt) g*(År) = ˚g(zr); i.e., zt zr. So, if ℋ: XW Z is defined by ℋ(År) = zr, where zr is the wpoint in Z induced by År, then ℋ is welldefined and oneone. For any z Z, let ˚Åz be the basic open Dfilter at z Z generated by ˚Vz = {°f°H˚f−1((˚f(z) − , ˚f(z) + )) ˚H [˚D] , > 0}. Since h is oneone, f = ˚f o h and h(X) is dense in Z, so h(fHf−1((˚f(z) − , ˚f(z) + ))) = °f°H˚f−1 C C Copyright © 2012 SciRes. APM
H. J. WU, W.H. WU opyrigh2012 SciRes. APM 300 Ct © ((˚f(z) − , ˚f(z) + ))) h(X) for any H [D]< , ˚H = {˚ff H}, > 0. Thus Vz = {fHf−1((˚f(z) − , ˚f(z) + )) fHf−1((˚f(z) − , ˚f(z) + )) for any H [D]< , > 0} is a welldefined open D Cfilter base on X. Let Åz be the basic open D Cfilter on X generated by Vz. If zo is the wpoint in Z induced by Åz. Then ˚f(zo) = ˚f(z) = f*(Åz) for all ˚f ˚D and f* D*. This implies that z = zo in Z. So, for any z Z, there is a unique Åz in XW such that ℋ(Åz) = z. Hence, ℋ is welldefined, oneone and onto. Theorem 5.27 (XW, k) is homeomorphic to (Z, h) under the mapping ℋ such that ℋ(k(x)) = h(x). Proof. Since the topologies on Z and XW are the weak topologies induced by ˚D and D*, respectively, to show the continuity of ℋ, it is enough to show that for any ˚f ˚D (or f* D*), any > 0, ℋ−1(˚f−1((tf − , tf + ))) = f*−1((tf − , tf + )). For each Ås in XW, let Vs = {fHf−1((sf − , sf + ))fHf−1((sf − , sf + )) for any H [D]< , > 0} be the open D Cfilter base on X that generates Ås. Let zs be the wpoint in Z induced by Ås, then ˚f(zs) = sf = f*(Ås). Thus (a): [Ås f*−1((tf − , tf + ))] iff (b): [˚f(zs) = f*(Ås) = sf (tf − , tf + )]. Since ℋ(Ås) = zs, so (b) iff (c): [ℋ(Ås) = zs ˚f−1((tf − , tf + ))] and (c) iff (d): [Ås ℋ−1(˚f−1((tf − , tf + )))]; i.e., f*−1((tf − , tf + )) = ℋ−1(˚f−1((tf − , tf + ))). So, ℋ is continuous. Since ℋ is oneone, onto and Z, XW are compact Haus dorff, by Theorem 17.14 in the Ref. [1, p. 123], ℋ is a homeomorphism. For that ℋ(k(x)) = h(x) is obvious from the definitions of k and h. Corollary 5.28 Let (X, h) be the StoneČech compactification of a Tychonoff space X, D = {ff = ˚f o h, ˚f C(X)} and ℋ:XW X is defined similarly to ℋ as above. Then (X, h) is homeomorphic to (XW, k) such that ℋ(k(x)) = h(x). Corollary 5.29 Let (X, h) be the Wallman compactifi cation of a normal T1space X, D = {ff = ˚f o h, ˚f C(X)} and ℋ:XW X is defined similarly to ℋ as above. Then (X, h) is homeomorphic to (XW, k) such that ℋ(k(x)) = h(x). REFERENCES [1] S. Willard, “General Topology,” AddisonWesley, Read ing, 1970. [2] H. J. Wu and W. H. Wu, “An Arbitrary Hausdorff Com pactification of a Tychonoff Space X Obtained from C Base by a Modified Wallman Method,” Topology and its Applications, Vol. 155, 2008, pp. 11631168. doi:10.1016/j.topol.2007.05.021 [3] J. L. Kelly, “General Topology,” Van Nostrand, Princeton, 1955.
