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

#### 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 no -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 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 -spaces introduced in [4], that is, a space is linearly 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 -spaces.

#### 2. Definitions

Note that throughout the paper, we denote by the family and by the set for any set and any family of a space . In particular, if , then we use the symbols and instead of and .

To make it easier to read, we recall some definitions.

A family of subsets of a space is *star-finite (star-countable)* if is finite (countable) for every .

A space is *strongly paracompact* if every open cover of has a star-finite open refinement.

A family of subsets of a space is *locally finite* if each has a neighborhood meeting only finitely many .

A space is *paracompact* if every open cover of has a locally finite open refinement.

A space is *countably paracompact* if every countable open cover of has a locally finite open refinement.

A space is *perfectly normal* if each pear of disjoint closed sets and in , there is a continuous function such that and . Here, the space is the open interval of reals equipped with usual metric topology.

A subset of a space is *discrete* if each has a neighborhood meeting at most one element in .

The *extent* of a space is the smallest infinite cardinal number such that for every discrete subset of .

A space is *linearly Lindelöf* if every increasing open cover of has a countable subcover. In the paper, we call a family of subsets of is *increasing* if the family is well ordered by proper inclusion.

A space is *linearly * provided that every increasing open cover of without a countable subcover has a closed and discrete -big set. Here, a set is -big if for every . Note that in -spaces, every discrete subset is closed. So in the proof of Theorem 3.6, we only need to prove that the increasing open cover has a discrete -big set.

A space is *monotonically normal* if to each pair of disjoint closed subsets of , one can assign an open set such that(i);(ii)if and , then .

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 Theorem 3.1, 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 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 is a countably paracompact normal space and every increasing open cover of has a star-countable open refinement. To prove that is strongly paracompact, let be an increasing open cover of and suppose that is a star-countable open refinement of . With the help of Lemma 3.2, we prove that the cover has a star-finite open refinement.

Firstly, we present the family in the following way.*Claim.* The family can be presented as , where each is a countable family and for .*Proof of claim.* For all , we call the finite subfamily a chain from to , if , , and for . For every , denote

It is easy to know that is countable, and, for any , if and only if . We complete the proof of the claim.

For every , let . By the above claim, we know that the family is an open and closed disjoint family of . Since is countably paracompact, the closed subspace of is countably paracompact for every . Moreover, it follows from the above claim that the family is a countable open cover of . By Lemma 3.3, we find a star-finite open family of the subspace refining . Since each is open in and since , it follows that the family is an open cover of . The family is also star-finite since is a disjoint family of . On the other hand, it is easy to see that is a refinement of since refines .

By Lemma 3.2, the space is strongly paracompact.

*Remark 3.4. *It is well known that space 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 Theorem 3.1, we obtain a characterization of linearly -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 is linearly provided that every increasing open cover of has a point-countable open refinement.*

*Proof. *Assume that is an increasing open cover of without a countable subcover, and is a point-countable open refinement of .

In order to prove easily, well order as and let . Since is point countable, the family is countable. For every , let be the first set of such that and let . The family cannot cover since is a countable family and has no countable subcover according to our assumption above. We then take the first point of which is not contained in and denote it by . For every , let be the first set of such that . The family is still not a cover of . Consequently, we are able to take the first point of which is not contained in and denote it by . Thus, we continue to define the family , where each is the first set of such that . Define and successively in the same way. There must exist an ordinal such that the set satisfies that the family covers .

To prove that the set is closed and discrete in , it suffices to show that is discrete since is . For every , if there exists some such that , let be the least such that . Then and for every with . On the other hand, for every , we know that and , where . It follows that . Thus we have proved that such a neighborhood of contains only one element of . By the arbitrariness of , we know that the set is discrete.

To prove is linearly , it is enough to show that is a -big set. To show this, pick an arbitrary . Assume on the contrary that for every . Then . It is contradicted with the fact that has no countable subcover. Therefore, there exists some such that . Then, we have . Thus we know that is a -big set.

We complete the proof of Theorem 3.6.

Since a space of countable extent is linearly Lindelöf if and only if it is linearly (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 linearly -space is paracompact.*

Theorem 3.9. *A monotonically normal space is strongly paracompact if and only if every increasing open cover of has a star-countable open refinement.*

*Proof. **Necessity.* It is trivial by the definition of a strongly paracompact space.*Sufficiency.* Assume that is a monotonically normal space and every increasing open cover of has a star-countable open refinement. It follows from Theorem 3.6 and Lemma 3.8 that is paracompact. By Theorem 3.1, we know that is strongly paracompact.

#### Acknowledgments

This paper was supported by Natural Science Foundation of China Grant 11026108 and Natural Science Foundation of Shandong Province Grants ZR2010AQ012, ZR2010AM019, and ZR2011AQ015.