A characterization of metric completeness

Weston, J. D.

doi:10.1090/S0002-9939-1977-0458359-2

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 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of metric completeness
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by J. D. Weston PDF

Proc. Amer. Math. Soc. 64 (1977), 186-188 Request permission

Abstract:

A proof is given of a theorem, relevant to fixed-point theory, which implies that a metric space (X, d) is complete if and only if, for each continuous function $h:X \to {\mathbf {R}}$ bounded below on X, there is a point ${x_0}$ such that $h({x_0}) - h(x) < d({x_0},x)$ for every other point x.

References

Chi Song Wong, On a fixed point theorem of contractive type, Proc. Amer. Math. Soc. 57 (1976), no. 2, 283–284. MR 407826, DOI 10.1090/S0002-9939-1976-0407826-5
B. Fisher, A fixed point theorem, Math. Mag. 48 (1975), no. 4, 223–225. MR 377842, DOI 10.2307/2690350