*Definition [#lede035e]
Let E be an ultrafilter of [[cozero set]]s 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 [#l37d7508]
-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 [#j8c1e621]
John J. Schommer and Mary Anne Swardson, ''Almost* realcompactness'', Commentationes Mathematicae Universitatis Carolinae, Vol.42 (2001), No.2, 383-392.