almost* realcompact

Definition

Let E be an ultrafilter of cozero sets and let F denote the collection of the closures of all elements in E. A Tychonoff space is called almost* realcompact if whenever E has ccip, F has a nonempty intersection.

Property

• If X is realcompact then X is almost* realcompact.
• A Tychonoff almost realcompact space is not necessarily almost* realcompact.
• Every almost* realcompact space is c-realcompact.
• Every almost* realcompact space is ψ-compact
• If X is super countably paracompact and almost realcompact then X is realcompact

Reference

John J. Schommer and Mary Anne Swardson, Almost* realcompactness, Commentationes Mathematicae Universitatis Carolinae, Vol.42 (2001), No.2, 383-392.