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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 186-188
  • MSC: Primary 54C30
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0458359-2
  • MathSciNet review: 0458359