Filters, consonance and hereditary Baireness
type: Item
instance of: scholarly article
title: Filters, consonance and hereditary Baireness
zbMATH Open document ID: 0964.54005
DOI: 10.1016/S0166-8641(99)00014-0
published in: Topology and its Applications
publication date: 2001-07-19T00:00:00Z
review text: For a topological space \(X\), let \(F(X)\) be the family of all closed subsets of \(X\). For \(F(X)\) the co-compact topology \(\tau_{\text{co}}\) is the topology generated by the family of all subsets of the form \(\{F\in F(X):F\cap K=\emptyset\}\), where \(K\) is an arbitrary compact subset of \(X\), and the upper Kuratowski topology \(\tau_{uk}\) is the topology associated with the upper Kuratowski-Painlevé convergence, where a net \((F_\gamma)_{\gamma \in\Gamma}\subset F(X)\) upper Kuratowski-Painlevé converges to \(F\in F(X)\) means that \(\bigcap\{\overline {\bigcup\{F_\gamma: \gamma\geq \alpha}\}: \alpha\in \Gamma\}\subset F\). The space \(X\) is said to be consonant if the equality \(\tau_{uk}= \tau_{\text{co}}\) holds. For a filter \({\mathcal F}\) on the set of natural numbers \(\omega\), let \(X_{\mathcal F}=\omega \cup\{\infty\}\) be the space for which all points in \(\omega\) are isolated and the neighborhood system of \(\infty\) is \(\{A\cup\{\infty\}: A\in{\mathcal F}\}\). The author gives a combinatorial characterization of the class \(\Phi\) of all filters \({\mathcal F}\) such that the space \(X_{\mathcal F}\) is consonant and all its compact subsets are finite. It is also shown that a filter \({\mathcal F}\) belongs to \(\Phi\) if and only if the space \(C_p(X_{\mathcal F})\) of real-valued continuous functions on \(X_{\mathcal F}\) with the pointwise-convergence topology is hereditarily Baire. Let \(\beta\omega\) denote the Čech-Stone compactification of the discrete space \(\omega\) and \(\omega^*= \beta\omega \smallsetminus\omega\). For a filter \({\mathcal F}\) on \(\omega\), let \(A({\mathcal F})\) be the subset of \(\omega^*\) of all ultrafilters on \(\omega\) finer than \({\mathcal F}\). \({\mathcal F}\) is said to be a \(P\)-filter if \(A({\mathcal F})\) is a \(P\)-set of \(\omega^*\); equivalently, for every sequence \((A_n)_{n\in \omega}\subset {\mathcal F}\) there is \(A\in{\mathcal F}\) such that \(A\subseteq^*A\) (that is, \(A\small…
reviewed by: Akihiro Okuyama
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 1462691
zbMATH Keywords: co-compact topology
zbMATH Keywords: Kuratowski topology
zbMATH Keywords: Cantor set
zbMATH Keywords: hereditarily Baire
author: Ahmed Bouziad
MaRDI profile type: Publication
full work available at URL: s0166-8641(99)00014-0
OpenAlex ID: W2047089896
description: scientific article; zbMATH DE number 1462691