## On strongly compact topological spaces.(English)Zbl 0647.54018

This paper is mainly concerned with alternative characterizations of strongly compact spaces. A subset S of a topological space (X,$${\mathcal T})$$ is called preopen if $$S\subset int(cl S)$$. A space (X,$${\mathcal T})$$ is strongly compact if every preopen cover of (X,$${\mathcal T})$$ has a finite subcover. The authors prove that a space (X,$${\mathcal T})$$ is strongly compact iff it is compact and satisfies one of the following conditions: (1) Every cover of X by dense subsets has a finite subcover. (2) Each set with empty interior in (X,$${\mathcal T})$$ is finite. (3) The family of dense sets in (X,$${\mathcal T})$$ is finite. (4) The set of all non-isolated points in (X,$${\mathcal T})$$ is finite. Finally they raise two questions, one of which is related to an SI-space in the sense of E. Hewitt [Duke Math. J. 10, 309-333 (1943; Zbl 0060.394)].
Reviewer: T.Ishii

### MSC:

 54D30 Compactness 54A05 Topological spaces and generalizations (closure spaces, etc.)

### Keywords:

strongly compact spaces; preopen cover; SI-space

Zbl 0060.394