almost-compact

Definition

Let X be a Hausdorff space, let E be a family of open sets in X and let F denote the family of the closures of all elements in E. X is called almost-compact if F has a nonempty intersection whenever E has fip.

Property

• If X is a Hausdorff space, then almost-compactness is equivalent to the following condition: if C is a open cover of X, there is a finite subcover F of C such that the closures of the member of F cover X.
• The product of nonempty Hausdorff space is almost-compact iff each coordinate space is almost-compact (See almost precompact and [Fletcher-Naimpally]).

Remark

• It is called H-closed in the terminology of M. Kateiov.

Reference

• Zdenek Frolik, A generalization of realcompact spaces, Czechoslovak Mathematical Journal, Vol.13 (1963), No. 1, 127-138.
• P. Fletcher and S. Naimpally, On almost complete and almost precompact quasi-uniform spaces, Czechoslovak Math. J., Vol.21 (1971), No.3, pp.383-390.