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ONE-POINT EXTENSIONS AND LOCAL TOPOLOGICAL PROPERTIES

Part of: Fairly general properties

Published online by Cambridge University Press:  02 August 2012

M. R. KOUSHESH
M. R. KOUSHESH*
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran (email: koushesh@cc.iut.ac.ir)
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Abstract

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A space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\setminus X$ is a singleton. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, Mrówka and Tsai [‘On local topological properties. II’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.19 (1971), 1035–1040] posed the following more general question: For what pairs of topological properties ${\mathscr P}$ and ${\mathscr Q}$ does a locally-${\mathscr P}$ space $X$ having ${\mathscr Q}$ possess a one-point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this old question.

Keywords

Stone–Čech compactificationone-point extensionone-point compactificationMrówka’s condition W

MSC classification

Secondary: 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 12 - 16
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

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