Extension properties induced by complete quasi-uniformities
type: Item
instance of: scholarly article
title: Extension properties induced by complete quasi-uniformities
zbMATH Open document ID: 0531.54035
author: P. Fletcher
author: Hans Peter Künzi
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 3842725
zbMATH Keywords: almost precompactness
zbMATH Keywords: fine quasi-uniformity of \(\Sigma\)-product with compact factors
zbMATH Keywords: completeness
zbMATH Keywords: Pervin quasi-proximity class
zbMATH Keywords: extension properties
zbMATH Keywords: almost realcompactness
zbMATH Keywords: countably paracompact space of non- measurable cardinality
zbMATH Keywords: maximal P-extension
zbMATH Keywords: FINE-complete space
zbMATH Keywords: Souslin property
MaRDI profile type: Publication
full work available at URL: pjm.1985.120.357
OpenAlex ID: W2029790914
DOI: 10.2140/PJM.1985.120.357
description: scientific article; zbMATH DE number 3842725