# Characterizations of Strongly Paracompact Spaces

Authors: Xin Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Characterizations of strongly compact spaces are given based on the existence of a star-countable open refinement for every increasing open cover. It is proved that a countably paracompact normal space (a perfectly normal space or a monotonically normal space) is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement. Moreover, it is shown that a space is linearlyD provided that every increasing open cover of the space has a point-countable open refinement.


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## Body

## 1. Introduction

The strongly paracompact property has been an interesting covering property in general topology. It is a natural generalization of compact spaces. It retains enough structure to enjoy many of the properties of compact spaces, yet sufficiently general to include a much wider class of spaces. On one hand, the strongly paracompact property is special since it is different in many aspects with other covering properties. For example, it is not implied even by metrizability; it is not preserved under finite-to-one closed mappings; it has noFσ-heredity. On the other hand, the property is general since every regular Lindelöf space is strongly paracompact.Unlike paracompactness, the strongly paracompact property has not many characterizations. The definition of the property is based on the existence of star-finite open refinement of every open cover. It is difficult to discover strongly paracompact spaces with only such a definition. So it has been an interesting subject to characterize the class in easier ways. In [1], Smirnov characterized the class in the way that a regular space is strongly paracompact if and only if every open cover of the space has a star-countable open refinement. Recently, Qu showed us another characterization in [2] that a regular space is strongly paracompact if and only if every increasing open cover of the space has a star-finite open refinement. The Tychonoff linearly Lindelöf nonparacompact space constructed in [3] helps us to know that we cannot obtain the conclusion only by weakening the condition “star-finite” in Qu’s result to “star-countable.” Then, it is natural to consider what more conditions we need to characterize the strongly paracompact space in the way that every increasing open cover of the space has a star-countable open refinement.In Section2, we mainly deal with this problem and first obtain that a countably paracompact normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement. Moreover, we also obtain a characterization of linearly D-spaces introduced in [4], that is, a space is linearly D provided that every increasing open cover of the space has a point-countable open refinement. It helps us to know that a monotonically normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement.Throughout the paper, all spaces are assumed to be regularT1-spaces.


## 2. Definitions

Note that throughout the paper, we denote by(ℱ)A the family {F∈ℱ:F∩A≠∅} and by St(A;ℱ) the set ⋃{F∈ℱ:F∩A≠∅} for any set A and any family ℱ of a space X. In particular, if A={x}, then we use the symbols (ℱ)x and St(x;ℱ) instead of (ℱ){x} and St({x};ℱ).To make it easier to read, we recall some definitions.A family𝒜 of subsets of a space is star-finite (star-countable) if (𝒜)A is finite (countable) for every A∈𝒜.A spaceX is strongly paracompact if every open cover of X has a star-finite open refinement.A family𝒜 of subsets of a space X is locally finite if each x∈X has a neighborhood meeting only finitely many A∈𝒜.A spaceX is paracompact if every open cover of X has a locally finite open refinement.A spaceX is countably paracompact if every countable open cover of X has a locally finite open refinement.A spaceX is perfectly normal if each pear of disjoint closed sets A and B in X, there is a continuous function f:X→𝕀 such that A=f-1(0) and A=f-1(1). Here, the space 𝕀 is the open interval (0,1) of reals equipped with usual metric topology.A subsetB of a space X is discrete if each x∈X has a neighborhood meeting at most one element in B.Theextent of a space X is the smallest infinite cardinal number τ such that |F|≤τ for every discrete subset F of X.A spaceX is linearly Lindelöf if every increasing open cover of X has a countable subcover. In the paper, we call a family 𝒪 of subsets of X is increasing if the family is well ordered by proper inclusion.A spaceX is linearlyD provided that every increasing open cover 𝒰 of X without a countable subcover has a closed and discrete 𝒰-big set. Here, a set A is 𝒰-big if A⊈U for every U∈𝒰. Note that in T1-spaces, every discrete subset is closed. So in the proof of Theorem 3.6, we only need to prove that the increasing open cover 𝒰 has a discrete 𝒰-big set.A spaceX is monotonically normal if to each pair (H,K) of disjoint closed subsets of X, one can assign an open set D(H,K) such that(i)
H⊂D(H,K)⊂D(H,K)¯⊂X∖K;(ii)
ifH⊂H′ and K⊃K′, then D(H,K)⊂D(H′,K′).For terminologies without definitions that appear in the paper, we refer the readers to [5, 6].


## 3. Main Results

Theorem 3.1.
A countably paracompact normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement.In order to prove Theorem3.1, we need the following results.Lemma 3.2 (see [2]).
A space is strongly paracompact if and only if every increasing open cover of the space has a star-finite open refinement.Lemma 3.3 (see [7]).
Every countable open cover of a countably paracompact normal space has a star-finite open refinement.Proof of Theorem3.1.
Necessity. By the definition of a strongly paracompact space, it is trivial to know that every increasing open cover of the space has a star-countable open refinement.
Sufficiency. Assume that X is a countably paracompact normal space and every increasing open cover of X has a star-countable open refinement. To prove that X is strongly paracompact, let 𝒪 be an increasing open cover of X and suppose that 𝒰 is a star-countable open refinement of 𝒪. With the help of Lemma 3.2, we prove that the cover 𝒪 has a star-finite open refinement.
Firstly, we present the family𝒰 in the following way.
Claim. The family 𝒰 can be presented as 𝒰={ℬα:α∈Λ}, where each ℬα is a countable family and (⋃ℬα)⋂(⋃ℬβ)=∅ for α≠β.
Proof of claim. For all A,B∈𝒰, we call the finite subfamily {C1,C2,…,Cn} a chain from A to B, if A=C1, B=Cn, and Ci∩Ci+1≠∅ for 1≤i<n. For every A∈𝒰, denote
(3.1)ℬ(A)={B∈𝒰:thereisachainfromAtoB}.
It is easy to know thatℬ(A) is countable, and, for any A1,A2∈𝒰, (⋃ℬ(A1))⋂(⋃ℬ(A2))≠∅ if and only if ℬ(A1)=ℬ(A2). We complete the proof of the claim.
For everyα∈Λ, let Zα=∪ℬα. By the above claim, we know that the family {Zα:α∈Λ} is an open and closed disjoint family of X. Since X is countably paracompact, the closed subspace Zα of X is countably paracompact for every α∈Λ. Moreover, it follows from the above claim that the family ℬα is a countable open cover of Zα. By Lemma 3.3, we find a star-finite open family 𝒲α of the subspace Zα refining ℬα. Since each Zα is open in X and since X=⋃α∈ΛZα, it follows that the family ⋃α∈Λ𝒲α is an open cover of X. The family ⋃α∈Λ𝒲α is also star-finite since {Zα:α∈Λ} is a disjoint family of X. On the other hand, it is easy to see that ⋃α∈Λ𝒲α is a refinement of 𝒪 since 𝒰 refines 𝒪.
By Lemma3.2, the space X is strongly paracompact.Remark 3.4.
It is well known that spaceΓ constructed in [3] is not strongly paracompact, while every increasing open cover of the space has a star-countable refinement since it is linearly Lindelöf. It helps us to know that in Theorem 3.1 we cannot get the conclusion if we remove the countably paracompact property.Corollary 3.5.
Every perfectly normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement.Proof.
Necessity. It is trivial by the definition of strongly paracompact spaces.
Sufficiency. It is known that every perfectly normal space is countably paracompact and normal (see [5]). Then it follows from Theorem 3.1 that a perfectly normal space is strongly paracompact if every increasing open cover of the space has a star-countable open refinement.Motivated by Theorem3.1, we obtain a characterization of linearly D-spaces in the way that every open cover of the space has a point-countable open refinement, which will help us to obtain a new characterization of strongly paracompact spaces in monotonically normal spaces.Theorem 3.6.
A spaceX is linearly D provided that every increasing open cover of X has a point-countable open refinement.Proof.
Assume that𝒰 is an increasing open cover of X without a countable subcover, and 𝒱 is a point-countable open refinement of 𝒰.
In order to prove easily, well orderX as {xα:α<Γ} and let y0=x0. Since 𝒱 is point countable, the family (𝒱)y0 is countable. For every V∈(𝒱)y0, let UV be the first set of 𝒰 such that UV⊃V and let 𝒰0={UV∈𝒰:V∈(𝒱)y0}. The family 𝒰0 cannot cover X since 𝒰0 is a countable family and 𝒰 has no countable subcover according to our assumption above. We then take the first point of X which is not contained in ∪𝒰0 and denote it by y1. For every V∈(𝒱)y1, let UV be the first set of 𝒰 such that V⊂UV. The family 𝒰1={UV∈𝒰:V∈(𝒱){y0,y1}} is still not a cover of X. Consequently, we are able to take the first point of X which is not contained in ∪𝒰1 and denote it by y2. Thus, we continue to define the family 𝒰2={UV∈𝒰:V∈𝒱{y0,y1,y2}}, where each UV is the first set of 𝒰 such that UV⊃V. Define yα and 𝒰α successively in the same way. There must exist an ordinal Λ≤Γ such that the set A={yα:α<Λ} satisfies that the family 𝒰′={UV∈𝒰:V∈(𝒱)A} covers X.
To prove that the setA is closed and discrete in X, it suffices to show that A is discrete since X is T1. For every x∈X, if there exists some V′∈(𝒱)x such that V′∩A≠∅, let β<Λ be the least such that yβ∈V′∩A. Then V′∈(𝒱)yβ and yα∉V′ for every yα with α<β. On the other hand, for every α>β, we know that yα∉∪𝒰β and V′∈𝒰β, where 𝒰β={UV∈𝒰:V∈(𝒱){yδ:δ≤β}}. It follows that yα∉V′. Thus we have proved that such a neighborhood V′ of x contains only one element of A. By the arbitrariness of x, we know that the set A is discrete.
To proveX is linearly D, it is enough to show that A is a 𝒰-big set. To show this, pick an arbitrary U∈𝒰. Assume on the contrary that U′⊂U for every U′∈𝒰′. Then ∪𝒰′⊂U. It is contradicted with the fact that 𝒰 has no countable subcover. Therefore, there exists some U′∈𝒰′ such that U⊂U′. Then, we have A⊄U. Thus we know that A is a 𝒰-big set.
We complete the proof of Theorem3.6.Since a space of countable extent is linearly Lindelöf if and only if it is linearlyD (see [4]), we have the following consequence of Theorem 3.6.Corollary 3.7.
A space of countable extent is linearly Lindelöf if and only if every increasing open cover of the space has a point-countable open refinement.At last, we close the paper with another main result with the help of foregoing results and the following lemma.Lemma 3.8 (see [4]).
Every monotonically normal linearlyD-space is paracompact.Theorem 3.9.
A monotonically normal spaceX is strongly paracompact if and only if every increasing open cover of X has a star-countable open refinement.Proof.
Necessity. It is trivial by the definition of a strongly paracompact space.
Sufficiency. Assume that X is a monotonically normal space and every increasing open cover of X has a star-countable open refinement. It follows from Theorem 3.6 and Lemma 3.8 that X is paracompact. By Theorem 3.1, we know that X is strongly paracompact.



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*Source: 916765-2012-12-23.xml*