On generalization of \(G_{\delta}\) diagonal and metrization
type: Item
instance of: scholarly article
title: On generalization of \(G_{\delta}\) diagonal and metrization
zbMATH Open document ID: 0678.54019
author: G. R. Hiremath
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: A cover of a topological space X is called semi-open if \(x\in Int St(x,{\mathbf{U}})\) for \(x\in X\), where St(x,\textbf{U})\(=\cup \{U\in {\mathbf{U}}|\) \(x\in X\}\). A topological space X is said to have \((\alpha_ 1)\) (resp. \((\alpha_ 2))\) if there is a sequence \(\{\) \textbf{U}\({}_ n\}_ n\) of semi-open covers with \(\cap_ nSt(x,{\mathbf{U}}_ n)=\{x\}\) (resp. \(\cup_ n\overline{St(x,{\mathbf{U}}_ n)}=\{x\})\) for each \(x\in X\). This paper studies topological spaces with \((\alpha_ i)\) in connection with the \(G_{\delta}\)-diagonal property defined by \textit{J. Chaber} [Bull. Acad. Pol. Sci., Ser. Sci. Math. 24, 993-998 (1976; Zbl 0347.54013)]. Both \((\alpha_ 2)\) and \(G_{\delta}\) diagonal property imply \((\alpha_ 1)\), but there is a hereditary paracompact space with \((\alpha_ 1)\) which does not have \(G_{\delta}\) diagonal property. An example shows that a \(T_ 2\) compact space satisfying \((\alpha_ 1)\) need not to be metrizable. It is known [cf. \textit{D. Burke}, General Topol. Appl. 2, 287-291 (1972; Zbl 0243.54017)] that \(G_{\delta}\) diagonal property is not preserved under perfect maps. This paper shows that \((\alpha_ 1)\) is preserved under a finite-to-one pseudo-open map. The author then uses this result to show that \(T_ 1\)-semidevelopability is preserved under a continuous finite-to-one pseudo-open maps.
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E30
zbMATH DE Number: 4111562
zbMATH Keywords: semi-open covers
zbMATH Keywords: \(G_{\delta }\)-diagonal property
zbMATH Keywords: hereditary paracompact space
zbMATH Keywords: \(T_ 2\) compact space
zbMATH Keywords: perfect maps
zbMATH Keywords: finite-to-one pseudo-open map
zbMATH Keywords: \(T_ 1\)-semidevelopability
MaRDI profile type: Publication
full work available at URL: pjaa.64.288
OpenAlex ID: W1965310103
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DOI: 10.3792/PJAA.64.288
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description: scientific article; zbMATH DE number 4111562