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Proceedings of the American Mathematical Society

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

 

 

Perfect maps of symmetrizable spaces


Author: Harold W. Martin
Journal: Proc. Amer. Math. Soc. 38 (1973), 410-412
MSC: Primary 54E35
DOI: https://doi.org/10.1090/S0002-9939-1973-0314009-3
MathSciNet review: 0314009
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Abstract: It is shown that if $ f:X \to Y$ is a perfect map from a symmetrizable space $ X$ onto a space $ Y$, then $ Y$ is metrizable if and only if $ f$ is a coherent map. This fact, together with certain known results, yields the following: Let $ f:X \to Y$ be a perfect map from a Hausdorff symmetrizable space $ X$ onto a space $ Y$; the following are equivalent: (1) $ X$ is metrizable; (2) $ f$ is a regular map; (3) $ f$ is a coherent map; (4) $ Y$ is metrizable.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0314009-3
Keywords: Metrizable space, symmetrizable space, symmetric, $ {G_\delta }$-diagonal, perfect map, coherent map, regular map
Article copyright: © Copyright 1973 American Mathematical Society