Bin, Zhao

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Inverse Limits of Spaces with the Weak B-Property

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Mathematical Journal of OkayamaUniversityVolume 50, Issue 1 2008 Article 6JANUARY 2008Inverse Limits of Spaces with the WeakB-PropertyZhao Bin∗∗Kashgar Teachers CollegeCopyright c©2008 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouMath. J. Okayama Univ. 50 (2008), 127–133INVERSE LIMITS OF SPACES WITH THE WEAKB-PROPERTYZhao BINAbstract. In this paper we show the following facts. Let X be theinverse limit space of an inverse system {Xα, piαβ , Λ}, where Λ is a di-rected set and |Λ| = κ > ω. Suppose that each projection piα : X → Xαis pseudo-open and X is κ-paracompact. If each Xα has the weak B-property, then X has the weak B-property. We also show that theanalogous result holds for hereditarily weak B-property.Recently, on the studies of products of normality or other covering prop-erties, a series of papers related to the following question proposed (see[1, 2, 3, 9, 10]).Question. For an inverse system {Xα, piαβ ,Λ} with |Λ| = κ > ω, supposethat the inverse limit space X of {Xα, piαβ ,Λ} is κ-paracompact and eachprojection piα : X → Xα is pseudo-open. If each Xα satisfies property P,then what property P can be preserved by the inverse limit space X?On the above question, the major work is by Chiba who shows that manycovering properties and some separation properties are preserved by theinverse limit space (see [1, 2, 3, 10]). The aim of this paper is to show thatthe above also holds under the weak B-property. About the property B, thesimilar results are announced in [1].The weak B-property as a generalization of the property B was first intro-duced by Yasui [7] who gave results relating to the property B and countableparacompactness. Subsequently many authors studied this property (see [6])and in 1985 Rudin [11] renamed this property D. However, in this paper weshall continue to use the term ”weak B-property”.Throughout this paper, we assume that all spaces are topological spaceswithout any separation axiom and all maps are continuous.Let X be a space and A ⊂ G ⊂ X, then A, intA denote the closure,interior of A in X respectively and AG, intG(A) denote the closure, interiorof A in G respectively. ω denotes the first infinite cardinal and [Σ]<ω denotesthe collection of all non-empty finite subsets of non-empty set Σ.Mathematics Subject Classification. Primary 54B10, 54D20; Secondary 54G20.Key words and phrases. inverse system, inverse limit space, weak B-property, heredi-tarily weak B-property.1271Bin: Inverse Limits of Spaces with the Weak B-PropertyProduced by The Berkeley Electronic Press, 2008128 Z. BINA map f from X onto Y is called pseudo-open if y ∈ intf(U) holds foreach y ∈ Y and each open set U in X with f−1(y) ⊂ U .It is easy to see that both open onto maps and closed onto maps arepseudo-open.Definition 1. Let κ be an arbitrary infinite cardinal. Then the space X iscalled κ-paracompact if every open cover U of X with |U| 6 κ has a locallyfinite open refinement.The following lemmas are necessary to prove our theorems.Lemma 1 ([9], Lemma 1.3). A space X is κ-paracompact if, and only if,for every open cover U = {Uα|α ∈ Λ} of X with |Λ| = κ, there is a locallyfinite open cover V = {Vα|α ∈ Λ} of X such that Vα ⊂ Uα for each α ∈ Λ.Lemma 2 ([2], Lemma 2). Let X be a κ-paracompact space, Λ a directedset with |Λ| = κ and U = {Uα|α ∈ Λ} an open cover of X such that Uα ⊂ Uβfor each α, β ∈ Λ with α 6 β. Then there is an open cover V = {Vα|α ∈ Λ}of X such that (i) Vα ⊂ Uα for each α ∈ Λ and (ii) Vα ⊂ Vβ if α 6 β.Let X be a κ-paracompact space and U = {Uα|α ∈ Λ} be an open coverof X satisfying the conditions of Lemma 2. Then U has an open refinementV = {Vα|α ∈ Λ} such that Vα ⊂ Uα (α ∈ Λ). By Lemma 1, V has a locallyfinite open refinement {Wα|α ∈ Λ} such that Wα ⊂ Vα (α ∈ Λ). Thereforewe have the following Lemma.Lemma 3 . Let X be a κ-paracompact space, Λ a directed set with |Λ| = κand U = {Uα|α ∈ Λ} an open cover of X such that Uα ⊂ Uβ for each α, β ∈Λ with α 6 β. Then there is a locally finite open cover V = {Vα|α ∈ Λ} ofX such that Vα ⊂ Uα for each α ∈ Λ.Definition 2. [10] Let κ be an arbitrary infinite cardinal. A space X hasthe property B∗(κ) if for any decreasing collection {Fα|α < κ} of closedsubsets in X with ∩{Fα|α < κ} = ∅, there exists a collection {Gα|α < κ} ofopen subsets of X such that Fα ⊂ Gα for each α < κ and ∩{Gα|α < κ} = ∅.A space X has the weak B-property if X has the property B∗(κ) for everyinfinite cardinal κ.It is easy to see that:Proposition 1. For a space X, the following are equivalent:2Mathematical Journal of Okayama University, Vol. 50 [2008], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol50/iss1/6INVERSE LIMITS OF SPACES WITH THE WEAK B-PROPERTY 129(1) X has the property B∗(κ).(2) [5] Every increasing open cover {Uα|α < κ} of X has an open cover{Vα|α < κ} of X such that Vα ⊂ Uα for each α < κ.Theorem 1. Let {Xα, piαβ ,Λ} be an inverse system and X be its inverselimit space lim←−{Xα, piαβ ,Λ}. Suppose that each projection piα : X → Xα is apsuedo-open map and X is a κ-paracompact space, where |Λ| = κ. If eachXα has the weak B-property, then X also has the weak B-property.Proof. Let τ be an arbitrary infinite cardinal and G = {Gξ |ξ < τ} bean increasing open cover of X. For each α ∈ Λ and ξ < τ , we put Uα,ξ =∪{U |Uopen in Xα, pi−1α (U) ⊂ Gξ} and Uα = ∪{Uα,ξ|ξ < τ}, then thecollection {pi−1α (Uα)|α ∈ Λ} satisfies:(1) ∪{pi−1α (Uα)|α ∈ Λ} = X.(2) pi−1α (Uα) ⊂ pi−1β (Uβ) if α 6 β.Since X is κ-paracompact, by Lemma 2, there is an open cover {Wα|α ∈Λ}, such that(3) Wα ⊂ pi−1α (Uα) for each α ∈ Λ.(4) If α 6 β, then Wα ⊂Wβ.Now, for each α ∈ Λ, we define the closed subset Tα = Xα\intpiα(X \Wα)of Xα. Since each projection piα : X → Xα is a psuedo-open map, we have(5) Tα ⊂ Uα for each α ∈ Λ.Put Cα = intpi−1α (Tα) for each α ∈ Λ, then(6) {Cα|α ∈ Λ} is an open cover of X.Because, for each x ∈ X, some Wα contains x. Hence there are β ∈ Λand an open subset V in Xβ such that x ∈ pi−1β (V ) ⊂ Wα. Then there isγ ∈ Λ with α, β 6 γ, and x ∈ pi−1β (V ) ⊂ pi−1γ (Tγ). Hence x ∈ Cγ .Since X is κ-paracompact, by Lemma 1, there is a locally finite open cover{Oα|α ∈ Λ} of X such that Oα ⊂ Cα for each α ∈ Λ. For each α ∈ Λ andξ < τ , we put U ′α,ξ = Uα,ξ ∪ (Xα \ Tα), then {U′α,ξ|ξ < τ} is an increasingopen cover of Xα since {Uα,ξ |ξ < τ} is an increasing open cover of Uα. Thuswe have(7) {Xα \ U′α,ξ|ξ < τ} is a decreasing collection of closed subsets of Xαsatisfying that ∩{Xα \ U′α,ξ|ξ < τ} = ∅.Since Xα has the weak B-property, there exists an open collection {Vα,ξ|ξ <τ} of Xα such that(8) Xα \ U′α,ξ ⊂ Vα,ξ for each ξ < τ .(9) ∩{Vα,ξ|ξ < τ} = ∅.3Bin: Inverse Limits of Spaces with the Weak B-PropertyProduced by The Berkeley Electronic Press, 2008130 Z. BINTo show that X has the weak B-property, by Proposition 1, it is sufficientto construct an open cover {Aξ |ξ < τ} of X satisfying Aξ ⊂ Gξ for eachξ < τ . Let Aξ = ∪{pi−1α (Xα \ Vα,ξ) ∩Oα|α ∈ Λ} for each ξ < τ . Then(10) {Aξ|ξ < τ} is an open cover of X.In the fact, for each x ∈ X, x ∈ Oα0 for some α0 ∈ Λ. Since piα0(x) ∈ Xα0and ∩{Vα0,ξ|ξ < τ} = ∅, we have xα0 ∈ Xα0 \ Vα0,ξ0 for some ξ0 < τ . Hencex ∈ pi−1α0 (Xα0 \ Vα0,ξ0) ∩Oα0 ⊂ Aξ0 .Last, we show that Aξ ⊂ Gξ for each ξ < τ . In the fact, for each ξ < τ ,observe that the collection {pi−1α (Xα \ Vα,ξ) ∩ Oα|α ∈ Λ} is locally finite inX. Therefore for each ξ < τ ,Aξ = ∪{pi−1α (Xα \ Vα,ξ) ∩Oα|α ∈ Λ}⊂ ∪{pi−1α (Xα \ Vα,ξ) ∩Oα|α ∈ Λ}= ∪{pi−1α (Xα \ Vα,ξ) ∩Oα|α ∈ Λ} ⊂ ∪{pi−1α (U′α,ξ) ∩Oα|α ∈ Λ}⊂ ∪{pi−1α (U′α,ξ) ∩ pi−1α (Tα)|α ∈ Λ}= ∪{pi−1α((Uα,ξ ∪ (Xα \ Tα)) ∩ Tα)|α ∈ Λ}= ∪{pi−1α (Uα,ξ ∩ Tα)|α ∈ Λ} ⊂ ∪{pi−1α (Uα,ξ)|α ∈ Λ} ⊂ GξThe proof of Theorem 1 is completed.We describe inverse limit spaces of the hereditarily weak B-propertiesA space X has the hereditarily weak B-property if every subspace of Xhas the weak B-property.It is not difficult to show the following lemma.Proposition 2. A space X has the hereditarily weak B-property i, and onlyif, every open subspace of X has the weak B-property.Theorem 2. Let {Xα, piαβ ,Λ} be an inverse system and X = lim←−{Xα, piαβ ,Λ}.Suppose that G is a κ−paracompact open subspace of X. If each Xα has thehereditarily weak B-property, then G has the weak B-property.Proof. Let τ be an arbitrary infinite cardinal and G = {Gξ |ξ < τ} be anincreasing open cover of G. For each α ∈ Λ and ξ < τ , we put Uα,ξ =∪{U |Uopen in Xα, pi−1α (U) ⊂ Gξ} and Uα = ∪{Uα,ξ|ξ < τ}, then similar tothe case of Theorem 1, the collection {pi−1α (Uα)|α ∈ Λ} satisfies(1) ∪{pi−1α (Uα)|α ∈ Λ} = G and(2) pi−1α (Uα) ⊂ pi−1β (Uβ) if α 6 β.Since G is a κ-paracompact open subspace of X and {pi−1α (Uα)|α ∈ Λ}satisfies the condition of Lemma 3, there is a locally finite open refinement4Mathematical Journal of Okayama University, Vol. 50 [2008], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol50/iss1/6INVERSE LIMITS OF SPACES WITH THE WEAK B-PROPERTY 131{Oα|α ∈ Λ} of {pi−1α (Uα)|α ∈ Λ} in G such that OαG⊂ pi−1α (Uα) for eachα ∈ Λ.Note that the collection Uα = {Uα,ξ|ξ < τ} is an increasing open cover ofUα because G is an increasing open cover of G. Therefore {Uα \Uα,ξ|ξ < τ}is a decreasing closed collection of Uα satisfying that ∩{Uα\Uα,ξ|ξ < τ} = ∅.Since Uα has the weak B-property, there exists an open collection {Vα,ξ|ξ <τ} in Uα such that(3) Uα \ Uα,ξ ⊂ Vα,ξ for each ξ < τ and(4) ∩{Vα,ξUα |ξ < τ}= ∅.To show that G has the weak B-property, by Proposition 1, it is sufficientto construct an open cover {Aξ|ξ < τ} of G satisfying AξG⊂ Gξ for eachξ < τ .Now, for each ξ < τ , we put Aξ = ∪{pi−1α (Uα \ Vα,ξUα) ∩ Oα| α ∈ Λ}.Then similar to the proof of Theorem 1, {Aξ|ξ < τ} is an open cover of G.Moreover we have(5) AξG⊂ Gξ for each ξ < τIndeed, for each ξ < τ , since the collection{pi−1α (Uα\Vα,ξUα)∩Oα| α ∈ Λ}is locally finite in G, we haveAξG= ∪{pi−1α (Uα \ Vα,ξUα) ∩OαG∣∣∣ α ∈ Λ}⊂ ∪{pi−1α (Uα \ Vα,ξ)G∩OαG∣∣ α ∈ Λ}⊂ ∪{pi−1α (Uα \ Vα,ξ) ∩G ∩ pi−1α (Uα)| α ∈ Λ}⊂ ∪{pi−1α (Uα \ Vα,ξ ∩ Uα) ∩G| α ∈ Λ}= ∪{pi−1α (Uα \ Vα,ξ) ∩G| α ∈ Λ} ⊂ ∪{pi−1α (Uα,ξ) ∩G| α ∈ Λ} ⊂ GξThe proof of Theorem 2 is completed.By Proposition 2 and Theorem 2, we obtain the following usual statementof inverse limits of the spaces with the hereditarily weak B-property.Corollary 1. Let {Xα, piαβ ,Λ} be an inverse system and X = lim←−{Xα, piαβ ,Λ}.Suppose that X is a hereditarily κ-paracompact space, where |Λ| = κ. If eachXα has the hereditarily weak B-property, then X also has the hereditarilyweak B-property.Finally, we study properties of the product of spaces with the weak B-properties.5Bin: Inverse Limits of Spaces with the Weak B-PropertyProduced by The Berkeley Electronic Press, 2008132 Z. BINLet κ be an infinite cardinal number, {Xα|α ∈ Σ} be a collection of spaceswith |Σ| = κ and X =∏α∈Σ Xα. We define the relation ” 6 ” of [Σ]<ω asA 6 B if and only if A ⊂ B for A,B ∈ [Σ]<ω and define the finite subprod-uct ZA =∏α∈A Xα of X for every A ∈ [Σ]<ω. Then [Σ]<ω is a directed setwith the relation ” 6 ”. For A 6 B (A,B ∈ [Σ]<ω), let piBA : ZB −→ ZAbe the natural projecion. Then piBA is an open bonding map from ZB ontoZA and hence we obtain the inverse system{ZA, piBA , [Σ]<ω}induced by thecollection {Xα|α ∈ Σ}, and by Lemma 1 of [2], every projection piA of thisinverse system{ZA, piBA , [Σ]<ω}is also an open map from the inverse limitspace Z = lim←−{ZA, piBA , [Σ]<ω}onto ZA.Lemma 4. [8] Suppose that {Xα|α ∈ Σ} is a collection of spaces, X =∏α∈Σ Xα and Z = lim←−{ZA, piBA , [Σ]<ω}, where |Σ| > ω. Then X and Z arehomeomorphic .Theorem 3. Suppose that {Xα|α ∈ Σ} is a collection of Hausdorff spacesand its product space X =∏α∈Σ Xα is κ-paracompact, where |Σ| = κ > ω.Then X has the weak B-property if, and only if, for each A ∈ [Σ]<ω thefinite subproduct ZA =∏α∈A Xα has the weak B-property.Proof. The ”only if” part follows from Lemma 4 and Theorem 1. For the”if” part, assume that X has the weak B-property. Then for each A ∈ [Σ]<ω,we choose a fixed point xα ∈ Xα for every α ∈ Σ \ A. Since ZA is homeo-morphic to the closed subspace∏α∈A Xα ×∏α∈Σ\A{xα} of X, ZA has theweak B-property.The proof of Theorem 3 is completed.Acknowledgments. The paper was written while the author is studyingin Chiba University as a foreign researcher. The author is most gratefulto professor Iwao Yoshioka for his many valuable comments and sugges-tions. The author is also most grateful to professor Masami Yasuda for hisencouragement and kind help.References[1] K. Chiba, Covering properties of inverse limits, Q & A in General Topology, Vol. 20(2002), 101-114[2] K. Chiba, Normality of inverse limits, Math. Japonica 35, No. 5 (1990), 959-970.[3] K. Chiba and Y. Yajima, Covering properties of inverse limits,II, Topology Proceed-ings, 27 (2003), 79-100[4] K. Chiba, On the weak B-property, Math. Japonica 29, No. 4 (1984), 551-567.6Mathematical Journal of Okayama University, Vol. 50 [2008], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol50/iss1/6INVERSE LIMITS OF SPACES WITH THE WEAK B-PROPERTY 133[5] Y. Yasui, On the characterization of the B-property by the normality of product spaces,Top. Appl. 15(1983), 323-326.[6] Y. Yasui, Generalized paracompactness, in: K. Morita and J. Nagata, Ed. Topics inGeneral Topology, North-Holland (1989), 161-202.[7] Y. Yasui, On the gaps between the refinements of the increasing open coverings, Proc.Japan Acad. 48(1972), 86-90.[8] R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag,Berlin, 1989.[9] Y. Aoki, Orthocompactness of inverse limits and products, Tsukuba J. Math., vol. 4(1980), 241-255.[10] K. Chiba, Strong Paracompactness and W-δθ-Refinability of Inverse Limits, Proc.Amer. Math. Soc. Vol. 134, No. 4 (2005), 1213-1221.[11] M. E. Rudin, κ-Dowker spaces, in London Math. Soc. Lecture Note Series 93(1985)175-195.Zhao BinDepartment of MathematicsKashgar Teachers CollegeKashgar, 844-007 Chinae-mail address: zhaob310@163.com(Received December 21, 2006 )7Bin: Inverse Limits of Spaces with the Weak B-PropertyProduced by The Berkeley Electronic Press, 2008

Inverse Limits of Spaces with the Weak B-Property

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