On certain Wallman spaces
type: Item
instance of: scholarly article
title: On certain Wallman spaces
zbMATH Open document ID: 0809.28009
DOI: 10.1155/S0161171294000396
author: Roy Connell
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1995-03-23T00:00:00Z
full work available at URL: https://eudml.org/doc/47106
review text: The purpose of the paper under review is to investigate the regularity of some Wallman spaces derived from sets of zero-one valued measures. More precisely: Let \(X\) be a non-empty set, let \(\mathcal L\) be a lattice of subsets of \(X\) containing \(\emptyset\) and \(X\), let \(A({\mathcal L})\) be the Boolean subalgebra of \(2^ X\) generated by \(\mathcal L\), let \(I({\mathcal L})= \{\mu: {\mathcal A}({\mathcal L})\to \{0,1\}\mid \mu\) is finitely additive\}, let \(I_ \sigma({\mathcal L})= \{\mu\in {\mathcal I}({\mathcal L})\): if \(L_ n\in {\mathcal L}\) and \(L_ n\downarrow\emptyset\), then \(\mu(L_ n)\to 0\}\) and let \(I^ \sigma({\mathcal L})= \{\mu\in {\mathcal I}({\mathcal L})\): \(\mu\) is countably additive\}. If \(I\) is a non-empty subset of \(I({\mathcal L})\), the Wallman topology on \(I\) is defined by taking the set \(\{\{\mu\in I: \mu(L)= 1\}: L\in {\mathcal L}\}\) as a base for the closed sets. Now the main results of the paper can be stated as follows: Theorem A. The Wallman space \(I_ \sigma({\mathcal L})\) is regular if and only if \(I_ \sigma({\mathcal L})= \{\mu\in I^ \sigma({\mathcal L})\): \(\mu\) is \({\mathcal L}\)-regular\}. Theorem B. The Walman space \(I^ \sigma({\mathcal L})\) is regular if and only if the following condition holds: For all \(\mu_ 1,\mu_ 2\in I({\mathcal L})\) and \(\nu\in I^ \sigma({\mathcal L})\), if \(\mu_ 1\leq \mu_ 2\) and \(\mu_ 1\leq \nu\) on \(\mathcal L\), then \(\mu_ 2\leq \nu\) on \(\mathcal L\).
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 28A60
zbMATH DE Number: 569394
zbMATH Keywords: \(\mathcal L\)-regular measures
zbMATH Keywords: regularity
zbMATH Keywords: Wallman spaces
zbMATH Keywords: zero-one valued measures
reviewed by: Pedro Morales
MaRDI profile type: Publication
OpenAlex ID: W2093713571
Recommended article: On extended Wallman type spaces
Recommended article: On Wallman Spaces and the Lindelöf Property
Recommended article: Q3779422
Recommended article: Q3469866
Recommended article: WALLMAN REMAINDERS OF REGULAR SPACES
Recommended article: THE WALLMAN COMPACTIFICATION AND INTERMEDIATE SPACES
Recommended article: Q5751574
Recommended article: On normal lattices and Wallman spaces
Recommended article: The Wall conjecture on domains in Euclidean spaces
Recommended article: On certain echelon spaces
description: scientific article; zbMATH DE number 569394