Monads and realcompactness
type: Item
instance of: scholarly article
title: Monads and realcompactness
zbMATH Open document ID: 0795.54044
DOI: 10.1016/0166-8641(94)90011-6
author: S. Salbany
author: Todor D. Todorov
published in: Topology and its Applications
publication date: 1994-04-18T00:00:00Z
full work available at URL: https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi
review text: A completely regular ordered topological space \((X,T,\leq)\) (in the sense of [\textit{L. Nachbin}, Topology and order (1965; Zbl 0131.379)]) is ordered real-compact if it is order isomorphic to a closed subspace of a product of copies of the usual ordered space \(\mathbb{R}\) of reals. The authors' main result is to give a ``quantifier free'' characterization of ordered realcompactness (which agrees with usual realcompactness when the order is discrete) in terms of A. Robinson-style nonstandard analysis. In particular, define a point \(\alpha\in{^*X}\) to be prenearstandard if \(^*f (\alpha)\) is finite in \(^*\mathbb{R}\) for every monotone nondecreasing continuous function \(f:X \to \mathbb{R}\). As usual, \(\alpha\) is nearstandard if there is some \(x \in X\) such that \(\alpha \in {^*U}\) for every open neighborhood \(U\) of \(x\). Clearly every nearstandard point of \(^*X\) is prenearstandard; the authors show that the converse holds if and only if \(X\) is ordered realcompact.
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54J05
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 54D60
zbMATH DE Number: 550033
zbMATH Keywords: ordered realcompactification
zbMATH Keywords: product space
zbMATH Keywords: nonstandard extension
zbMATH Keywords: nonstandard ordered hull
zbMATH Keywords: monads
reviewed by: Paul Bankston
MaRDI profile type: Publication
OpenAlex ID: W2050110458
cites work: Extensions of completely regular ordered spaces
cites work: C-complete quasi-uniform spaces
cites work: Q3270120
cites work: Q3708229
cites work: Q5340517
cites work: Nonstandard and standard compactifications of ordered topological spaces
description: scientific article; zbMATH DE number 550033