One-point extensions and local topological properties

Authors

  • M.R. Koushesh

Keywords:

Stone-Cech compactification, one-point extension, one-point compactification.

Abstract

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a {\em one--point extension} of $X$ if $Y\backslash X$ is a singleton. P. Alexandroff proved that any locally compact non--compact Hausdorff space $X$ has a one--point compact Hausdorff extension, called the {\em one--point compactification} of $X$. Motivated by this, S. Mr\'{o}wka and J.H. Tsai [On local topological properties. II, \emph{Bull. Acad. Polon. Sci. S\'{e}r. Sci. Math. Astronom. Phys.} {\bf 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 ${\mathscr Q}$ have a one--point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this question. 10.1017/S0004972712000524

Published

2013-07-23

Issue

Vol. 88 No. 1 (2013)

Section

Articles
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