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ONE-POINT EXTENSIONS AND LOCAL TOPOLOGICAL PROPERTIES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 1 / August 2013
- Published online by Cambridge University Press:
- 02 August 2012, pp. 12-16
- Print publication:
- August 2013
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- Article
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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.
