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Proceedings of the American Mathematical Society

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

 

 

Metrization of symmetric spaces and regular maps


Author: Harold W. Martin
Journal: Proc. Amer. Math. Soc. 35 (1972), 269-274
MSC: Primary 54E25
DOI: https://doi.org/10.1090/S0002-9939-1972-0303511-5
MathSciNet review: 0303511
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Abstract: A symmetric d for a topological space R is said to be coherent if whenever $ \{ x(n)\} $ and $ \{ y(n)\} $ are sequences in R with $ d(x(n),y(n)) \to 0$ and $ d(x(n),x) \to 0$, then $ d(y(n),x) \to 0$. V. Niemytzki and W. A. Wilson have essentially shown that a topological space R is metrizable if and only if R is symmetrizable via a coherent symmetric. Conditions on a symmetric d which are equivalent to d being coherent are established. As a consequence, a theorem of A. Arhangel'skiĭ may be refined by showing that if $ f:R \to Y$ is a quotient map from a metrizable space R onto a $ {T_0}$-space y, then Y is metrizable if and only if f is a regular map.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0303511-5
Keywords: Distance function, coherent distance function, distance space, symmetric, coherent symmetric, symmetrizable space, regular map, coherent map
Article copyright: © Copyright 1972 American Mathematical Society