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  • M. R. KOUSHESH (a1) (a2)

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.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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