functionally compact

Definition

• A Hausdorff topological space X is called functionally compact if whenever U is an open filter base on X such that the intersection A of the elements of U is equal to the intersection of the closures of the elements of U, then U is a base for the neighborhoods of A.

Property

• A functionally compact Hausdorff space is compact if and only if it is regular.
• A Hausdorff space X is functionally compact if and only if every mapping of X into any Hausdorff space is closed.
• If X is C-compact then X is functionally compact.
• Every functionally compact space is minimal Hausdorff.
• Every CFC space is FFC and every FFC space is FC.

Remark

• It is abbreviated as FC.
• See finitely functionally compact and compactly functionally compact.

Reference

• Dickman, R. F. and Jr. Zame, A.,Functionally compact spaces, Pacific J. Math. 31 1969 303--311
• R.F. Dickman, Jr. and J.R. Porter, Between minimal Hausdorff and compact Hausdorff spaces, Topology Proc. Vol.9 (1984), pp.243-268.