\(p\)-pseudocompactness and related topics in topological spaces
type: Item
instance of: scholarly article
title: \(p\)-pseudocompactness and related topics in topological spaces
zbMATH Open document ID: 0970.54008
DOI: 10.1016/S0166-8641(98)00111-4
published in: Topology and its Applications
publication date: 2001-10-16T00:00:00Z
review text: Let \(p\) be a non-principal ultrafilter on \(\omega\). A subset \(Y\) of a space \(X\) is called \(p\)-bounded in \(X\) if every sequence \(\{V_{n}: n\in\omega\}\) of open sets in \(X\) with \(V_{n}\cap Y\neq\emptyset\), for each \(n\in\omega\), has a \(p\)-limit point \(x\in X\). The property of being \(p\)-bounded is productive and preserved by continuous functions. In addition, if \(X\) is \(p\)-bounded for some \(p\in\omega^{*}\), then \(X\) is pseudocompact [\textit{S. Garcia-Ferreira}, Ann. N.Y. Acad. Sci. 728, 22-31 (1994; Zbl 0911.54022)]. The authors establish some new properties of \(p\)-bounded subsets formulated in terms of \(z\)-ultrafilters and families of equicontinuous functions. Then they analyze the relations of \(p\)-pseudocompactness with \(p\)-compactness and \(\alpha\)-pseudocompactness, \(\alpha\geq\omega\). It is proved that \(cl_{\mu(X\times Y)}(A\times B)=cl_{\mu{X}}(A) \times cl_{\mu{Y}}(B)\) for any \(p\)-bounded subsets \(A\subseteq X\) and \(B\subseteq Y\), where \(\mu\) denotes the Dieudonné completion of a space. An analog of Glicksberg's pseudocompact product theorem is proved for \(p\)-boundedness.
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 1367571
zbMATH Keywords: \(p\)-limit point
zbMATH Keywords: \(p\)-pseudocompact space
zbMATH Keywords: \(p\)-compact space
zbMATH Keywords: \(p\)-bounded set
zbMATH Keywords: \(C_{\alpha}\)-compact set
zbMATH Keywords: \(\alpha\)-pseudocompact space
zbMATH Keywords: degree of pseudocompactness
zbMATH Keywords: Glicksberg's theorem
zbMATH Keywords: Dieudonné completion
zbMATH Keywords: sequentially compact space
zbMATH Keywords: totally countably compact space
zbMATH Keywords: \(z\)-ultrafilter
author: Manuel Sanchis
author: Angel Tamariz-Mascarúa
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00111-4
OpenAlex ID: W2116574217
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description: scientific article; zbMATH DE number 1367571