Characterizations of Strongly Paracompact Spaces

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 linearly D provided that every increasing open cover of the space has a point-countable open refinement.


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 no F σheredity.On the other hand, the property is general since every regular Lindel öf 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 Lindel öf 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 Section 2, 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 regular T 1 -spaces.

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 family A of subsets of a space is star-finite (star-countable) if A A is finite countable for every A ∈ A.
A space X is strongly paracompact if every open cover of X has a star-finite open refinement.
A family A of subsets of a space X is locally finite if each x ∈ X has a neighborhood meeting only finitely many A ∈ A.
A space X is paracompact if every open cover of X has a locally finite open refinement.A space X is countably paracompact if every countable open cover of X has a locally finite open refinement.
A space X 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 subset B of a space X is discrete if each x ∈ X has a neighborhood meeting at most one element in B.
The extent of a space X is the smallest infinite cardinal number τ such that |F| ≤ τ for every discrete subset F of X.
A space X 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 space X is linearly D 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 T 1 -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 space X is monotonically normal if to each pair H, K of disjoint closed subsets of X, For terminologies without definitions that appear in the paper, we refer the readers to 5, 6 .

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 Theorem 3.1, we need the following results.
{B ∈ U : there is a chain from A to B}.

3.1
It is easy to know that B A is countable, and, for any We complete the proof of the claim.For every α ∈ Λ, let Z α ∪B α .By the above claim, we know that the family {Z α : α ∈ Λ} is an open and closed disjoint family of X.Since X is countably paracompact, the closed subspace Z α of X is countably paracompact for every α ∈ Λ.Moreover, it follows from the above claim that the family B α is a countable open cover of Z α .By Lemma 3.3, we find a starfinite open family W α of the subspace Z α refining B α .Since each Z α is open in X and since X α∈Λ Z α , it follows that the family α∈Λ W α is an open cover of X.The family α∈Λ W α is also star-finite since {Z α : α ∈ Λ} is a disjoint family of X.On the other hand, it is easy to see that α∈Λ W α is a refinement of O since U refines O.
By Lemma 3.2, the space X is strongly paracompact.
Remark 3.4.It is well known that space Γ constructed in 3 is not strongly paracompact, while every increasing open cover of the space has a star-countable refinement since it is linearly Lindel öf.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 Theorem 3.1, 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 space X is linearly D provided that every increasing open cover of X has a pointcountable open refinement.
Proof.Assume that U 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 order X as {x α : α < Γ} and let y 0 x 0 .Since V is point countable, the family V y 0 is countable.For every V ∈ V y 0 , let U V be the first set of U such that U V ⊃ V and let U 0 {U V ∈ U : V ∈ V y 0 }.The family U 0 cannot cover X since U 0 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 ∪U 0 and denote it by y 1 .For every V ∈ V y 1 , let U V be the first set of U such that V ⊂ U V .The family U 1 {U V ∈ U : V ∈ V {y 0 ,y 1 } } is still not a cover of X.Consequently, we are able to take the first point of X which is not contained in ∪U 1 and denote it by y 2 .Thus, we continue to define the family U 2 {U V ∈ U : V ∈ V {y 0 ,y 1 ,y 2 } }, where each U V is the first set of U such that U V ⊃ V .Define y α and U α successively in the same way.There must exist an ordinal Λ ≤ Γ such that the set A {y α : α < Λ} satisfies that the family U {U V ∈ U : V ∈ V A } covers X.
To prove that the set A is closed and discrete in X, it suffices to show that A is discrete since X is T 1 .For every x ∈ X, if there exists some V ∈ V x such that V ∩ A / ∅, let β < Λ be the least such that y β ∈ V ∩ A. Then V ∈ V y β and y α / ∈ V for every y α with α < β.On the other hand, for every α > β, we know that y α / ∈ ∪U β and V ∈ U β , where U β {U V ∈ U : V ∈ V {y δ :δ≤β} }.It follows that y α / ∈ 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 prove X 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 Theorem 3.6.
Since a space of countable extent is linearly Lindel öf if and only if it is linearly D 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.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.

Lemma 3 .8 see 4 .
Every monotonically normal linearly D-space is paracompact.Theorem 3.9.A monotonically normal space X 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.
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.Every countable open cover of a countably paracompact normal space has a star-finite open refinement.Proof of Theorem 3.1.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 family U in the following way.Claim.The family U can be presented as U {B Proof of claim.For all A, B ∈ U, we call the finite subfamily {C 1 , C 2 , . . ., C n } a chain from A to B, if A C 1 , B C n , and α : α ∈ Λ}, where each B α is a countable family andB α B β ∅ for α / β.