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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Topological completions of metrizable spaces


Authors: Ben Fitzpatrick, Gary F. Gruenhage and James W. Ott
Journal: Proc. Amer. Math. Soc. 117 (1993), 259-267
MSC: Primary 54E20; Secondary 54A25, 54D40, 54D45
DOI: https://doi.org/10.1090/S0002-9939-1993-1110542-8
MathSciNet review: 1110542
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Abstract: For a pair of metrizable spaces $ X$ and $ Y$, we investigate conditions under which there is a dense embedding $ h:X \to Z$, where $ Z$ is completely metrizable and $ Z\backslash h(X)$ is homeomorphic to $ Y$. In such a case, $ Z$ is called a topological completion of $ X$ and $ Y$ is called a completion remainder of $ X$. In case $ X$ and $ Y$ are completely metrizable, we give necessary and sufficient conditions that $ Y$ be a completion remainder of $ X$. We characterize the completion remainders of $ {\mathbf{R}}$ and those of the rationals, $ {\mathbf{Q}}$. We also characterize the remainders of $ {\mathbf{Q}}(\kappa )$, a nonseparable analogue of $ {\mathbf{Q}}$.


References [Enhancements On Off] (What's this?)

  • [1] Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
  • [2] S. V. Medvedev, Topological characteristics of spaces $ {\mathbf{Q}}(\kappa )$ and $ {\mathbf{Q}} \times B(\kappa )$, Moscow Univ. Math. Bull. 41 (1986), 42-45.
  • [3] R. L. Moore, A set of axioms for plane analysis situs, Fund. Math. 25 (1935), 13-28.
  • [4] V. Niemytzki and A. Tychonoff, Beweis des Satzes dass ein metrisierbarer Raum dann und nur dann kompact ist, wenn er in jeder Metrik volständig ist, Fund. Math. 12 (1928), 118-120.
  • [5] A. Wilansky and M. D. Mavinkurve, Problems and Solutions: Solutions of Advanced Problems: 5602, Amer. Math. Monthly 76 (1969), no. 5, 569. MR 1535451

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1110542-8
Keywords: Completely metrizable, topological completions, Polish spaces, $ \sigma $-compact, $ \sigma $-discrete, nowhere locally compact
Article copyright: © Copyright 1993 American Mathematical Society