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

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

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 [

In Section

Throughout the paper, all spaces are assumed to be regular

Note that throughout the paper, we denote by

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

A family

A space

A family

A space

A space

A space

A subset

The

A space

A space

A space

if

For terminologies without definitions that appear in the paper, we refer the readers to [

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

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.

Firstly, we present the family

It is easy to know that

For every

By Lemma

It is well known that space

Every perfectly normal space is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement.

Motivated by Theorem

A space

Assume that

In order to prove easily, well order

To prove that the set

To prove

We complete the proof of Theorem

Since a space of countable extent is linearly Lindelöf if and only if it is linearly

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.

Every monotonically normal linearly

A monotonically normal space

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