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almost precompact

Last-modified: 2010-08-28 (土) 05:26:37

Definition Edit

A quasi-uniform space X is almost precompact provided that if U is an entourage, there is a finite subset F of X such that X = cl(U[F]).

Property Edit

  • A quasi-uniform space is almost precompact iff every open ultrafilter on X is a Cauchy filter?.
  • A Hausdorff space X is almost-compact iff every compatible quasi-uniformity for X is almost complete and almost precompact.
  • (The Generalized Niemytzki-Tychonoff theorem) A Hausdorff space is almost-compact iff it is almost complete with respect to every compatible quasi-uniformity.
  • Let imgtex.fcgi?%5bres=100%5d%7b$%28X%2c%5cmathcal%7bU%7d%29$%7d%25.png be an almost precompact quasi-uniform space and let imgtex.fcgi?%5bres=100%5d%7b$%28X%5e*%2c%5cmathcal%7bU%7d%5e*%29$%7d%25.png be a regular completion of imgtex.fcgi?%5bres=100%5d%7b$%28X%2c%5cmathcal%7bU%7d%29$%7d%25.png. Then imgtex.fcgi?%5bres=100%5d%7b$%28X%5e*%2c%5cmathcal%7bU%7d%5e*%29$%7d%25.png is compact.

Remark Edit

Reference Edit

P. Fletcher and S. Naimpally, On almost complete and almost precompact quasi-uniform spaces, Czechoslovak Math. J., Vol.21 (1971), No.3, pp.383-390.