Definition
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
- 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 be an almost precompact quasi-uniform space and let be a regular completion of . Then is compact.
Remark
- See almost-compact.
Reference
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.