A metric characterization of zero-dimensional spaces

Janoš, Ludvík

doi:10.1090/S0002-9939-1972-0288739-5

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A metric characterization of zero-dimensional spaces
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by Ludvík Janoš

Proc. Amer. Math. Soc. 31 (1972), 268-270

DOI: https://doi.org/10.1090/S0002-9939-1972-0288739-5

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Abstract:

It is shown that a nonempty separable metrizable space $X$ is zero-dimensional if and only if there exists a metric $\rho$ on $X$, inducing the given topology of $X$ and such that all nonzero distances $\rho (x,y)$ are mutually different.

References

J. de Groot, On a metric that characterizes dimension, Canadian J. Math. 9 (1957), 511–514. MR 90804, DOI 10.4153/CJM-1957-059-6
Jun-iti Nagata, On a relation between dimension and metrization, Proc. Japan Acad. 32 (1956), 237–240. MR 86284

Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 268-270
DOI: https://doi.org/10.1090/S0002-9939-1972-0288739-5
MathSciNet review: 0288739