On certain Wallman spaces
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 \(\mathcal L\).
scientific article; zbMATH DE number 569394