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ONE-POINT EXTENSIONS AND LOCAL TOPOLOGICAL PROPERTIES

  • M. R. KOUSHESH (a1)
Abstract
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 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.

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References
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[1]Burke D. K., ‘Covering properties’, in: Handbook of Set-Theoretic Topology, (eds. Kunen K. & Vaughan J. E.) (Elsevier, Amsterdam, 1984), pp. 347422.
[2]Engelking R., General Topology, 2nd edn (Heldermann, Berlin, 1989).
[3]Engelking R. & Mrówka S., ‘On $E$-compact spaces’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958), 429436.
[4]Koushesh M. R., ‘Compactification-like extensions’, Dissertationes Math. (Rozprawy Mat.) 476 (2011), 88pp.
[5]Koushesh M. R., ‘The partially ordered set of one-point extensions’, Topology Appl. 158 (2011), 509532.
[6]Koushesh M. R., ‘A pseudocompactification’, Topology Appl. 158 (2011), 21912197.
[7]Mack J., Rayburn M. & Woods R. G., ‘Local topological properties and one-point extensions’, Canad. J. Math. 24 (1972), 338348.
[8]Mack J., Rayburn M. & Woods R. G., ‘Lattices of topological extensions’, Trans. Amer. Math. Soc. 189 (1974), 163174.
[9]Marin F. L., ‘A note on $E$-compact spaces’, Fund. Math. 76 (1972), 195206.
[10]Mrówka S., ‘On local topological properties’, Bull. Acad. Polon. Sci. 5 (1957), 951956.
[11]Mrówka S., ‘Further results on $E$-compact spaces. I’, Acta Math. 120 (1968), 161185.
[12]Mrówka S. & Tsai J. H., ‘On local topological properties. II’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 10351040.
[13]Stephenson R. M., ‘Initially $\kappa $-compact and related spaces’, in: Handbook of Set-Theoretic Topology, (eds. Kunen K. & Vaughan J. E.) (Elsevier, Amsterdam, 1984), pp. 603632.
[14]Tsai J. H., ‘More on local topological properties’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 4951.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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