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.