almost realcompact

Definition

A topological space X is said to be almost realcompact if every open ultrafilter U with the weak cip? has a nonempty adherence.

Property

• Every realcompact space is almost realcompact.
• Every normal almost realcompact space is realcompact.
• Almost realcompactness of a Tychonoff space X is equivalent to the following condition: The collection of all countable open covers of X is complete.
• Alomost realcompactness is invariant under perfect mappings.
• Conversely, let X be a regular space, Y an image of a perfect mapping. If Y is almost realcompact, X is also almost realcompact.
• Every Lindeloef space is almost realcompact.
• The topological product of an arbitrary family of almost realcompact spaces is an almost realcompact space.
• Closed subspaces of a regular almost realcompact space are almost realcompact.
  space.
• A Tychonoff space is almost realcompact iff every ultrafilter of regular closed sets with cip is fixed.

Remark

• cf. realcompact, almost* realcompact.
• Some authors require Tychonoffness ([Schommer, Swardson, 2001]).

Reference

• Zdenek Frolik, A generalization of realcompact spaces, Czechoslovak Mathematical Journal, Vol.13 (1963), No. 1, 127-138.
• John J. Schommer and Mary Anne Swardson, Almost realcompactness, Commentationes Mathematicae Universitatis Carolinae, Vol.2 (2001), No.2, 383-392.