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


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


Zbl 0060.394