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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
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?)

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Keywords: Completely metrizable, topological completions, Polish spaces, $ \sigma $-compact, $ \sigma $-discrete, nowhere locally compact
Article copyright: © Copyright 1993 American Mathematical Society

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