Extension properties induced by complete quasi-uniformities
We consider the completeness of the following members of the Pervin quasi-proximity class of a completely regular Hausdorff space: PF, LF, SC, FT and FINE. We show that these completeness properties are extension properties, as defined by R. G. Woods, which for PF, LF and SC are closely related to almost realcompactness. Indeed, in a countably paracompact space of non-measurable cardinality, PF-completeness, LF- completeness, SC-completeness and almost realcompactness coincide. We show that the fine quasi-uniformity of any \(\Sigma\)-product with compact factors is almost precompact, and it follows that no \(\Sigma\)-product is FINE-complete. If a \(\Sigma\)-product is \(C^*\)-embedded in its Tychonoff product \(\pi\), and if \(\pi\) is P-complete for any of the completeness properties under consideration, then \(\pi\) is the maximal P-extension of \(\Sigma\). The results concerning \(\Sigma\)-products rely on the following proposition: In a FINE-complete space with the Souslin property, the closure of each \(\omega\)-bounded subspace is compact.
fine quasi-uniformity of \(\Sigma\)-product with compact factors
countably paracompact space of non- measurable cardinality
scientific article; zbMATH DE number 3842725