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ONE-POINT CONNECTIFICATIONS

Part of: Basic constructions Fairly general properties

Published online by Cambridge University Press:  09 January 2015

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), PO 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 if $Y\setminus X$ is a singleton. Compact extensions are called compactifications and connected extensions are called connectifications. It is well known that every locally compact noncompact space has a one-point compactification (known as the Alexandroff compactification) obtained by adding a point at infinity. A locally connected disconnected space, however, may fail to have a one-point connectification. It is indeed a long-standing question of Alexandroff to characterize spaces which have a one-point connectification. Here we prove that in the class of completely regular spaces, a locally connected space has a one-point connectification if and only if it contains no compact component.

Keywords

Stone–Čech compactificationWallman compactificationone-point extensionone-point connectificationone-point compactificationlocally connected

MSC classification

Primary: 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Secondary: 54D05: Connected and locally connected spaces (general aspects) 54D40: Remainders 54B15: Quotient spaces, decompositions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 1 , August 2015 , pp. 76 - 84
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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