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almost realcompact

Last-modified: 2015-11-17 (火) 19:09:52

Definition Edit

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

Property Edit

  • 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.
  • A Tychonoff space is almost realcompact iff every ultrafilter of regular closed sets with cip is fixed.

Remark Edit

Reference Edit

  • 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.