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arXiv · 1308.5167

One-point connectifications

M. R. Koushesh
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Abstract

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-X is a singleton. Compact extensions are called compactifications and connected extensions are called connectifications. It is well known that every locally compact non-compact 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.

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M. R. Koushesh. 2014-10-30. One-point connectifications. https://doi.org/10.1017/s1446788714000676

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