Nearly realcompact spaces
type: Item
instance of: scholarly article
title: Nearly realcompact spaces
zbMATH Open document ID: 0772.54021
DOI: 10.1016/0166-8641(92)90031-T
author: Robert L. Blair
author: Eric K. van Douwen
published in: Topology and its Applications
publication date: 1993-04-01T00:00:00Z
review text: A Tychonoff space \(X\) is nearly realcompact if \(\beta X\setminus\upsilon X\) is dense in \(\beta X\setminus X\), where \(\beta X\) is the Stone- Čech compactification of \(X\) and \(\upsilon X\) is the Hewitt realcompactification of \(X\). While every realcompact space is obviously nearly realcompact, the converse does not hold. The authors obtain several characterizations for nearly realcompact spaces and also several sufficient conditions for a space to be nearly realcompact. Of basic importance in these results are the notions of local compactness and relative pseudocompactness. (A subset \(A\) of a space \(X\) is relatively pseudocompact in \(X\) if every continuous real-valued function on \(X\) is bounded on \(A\).) It is shown that the nearly realcompactness of a space \(X\) is equivalent to \(X\) containing the locally compact part of \(\upsilon X\) and also is equivalent to all relatively pseudocompact cozero-sets in \(X\) being \(\sigma\)-compact (or realcompact, or nearly realcompact). Among interesting nearly realcompact spaces are discrete spaces (with no measurable cardinal assumptions) and topologically complete spaces. Also, if the absolute of a space is nearly realcompact, so is the space; although the converse fails generally, it does hold if the space is nowhere locally compact. The authors also discuss product, sum, and mapping theorems, and several open questions are posed.
Mathematics Subject Classification ID: 54D60
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54C10
zbMATH DE Number: 149082
zbMATH Keywords: nearly realcompact spaces
zbMATH Keywords: local compactness
zbMATH Keywords: relative pseudocompactness
zbMATH Keywords: nearly realcompactness
reviewed by: Stephan C. Carlson
MaRDI profile type: Publication
cites work: Q4778037
cites work: Spaces in Which Special Sets are z-Embedded
cites work: Extensions of zero-sets and of real-valued functions
cites work: Spaces with an Oz Stone-Čech compactification
cites work: Q3038176
cites work: Mappings and realcompact spaces
cites work: Generalizations of realcompact spaces
cites work: Q3994935
cites work: Stone-Čech compactification of a product
cites work: On c-realcompact spaces and locally bounded normal functions
cites work: On nearly pseudocompact spaces
cites work: Measures in Fully Normal Spaces
cites work: On Countably Compact, Perfectly Normal Spaces
cites work: Q4074053
cites work: A Tychonoff Almost Realcompactification
cites work: Q3938873
description: scientific article; zbMATH DE number 149082