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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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Local isometries of compact metric spaces
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by Aleksander Całka PDF
Proc. Amer. Math. Soc. 85 (1982), 643-647 Request permission

Abstract:

By local isometries we mean mappings which locally preserve distances. A few of the main results are: 1. For each local isometry $f$ of a compact metric space $(M,\rho )$ into itself there exists a unique decomposition of $M$ into disjoint open sets, $M = M_0^f \cup \cdots \cup M_n^f$, $(0 \leqslant n < \infty )$ such that (i) $f(M_0^f) = M_0^f$, and (ii) $f(M_i^f) = M_{i - 1}^f$ and $M_i^f \ne \emptyset$ for each $i, 1 \leqslant i \leqslant n$. 2. Each local isometry of a metric continuum into itself is a homeomorphism onto itself. 3. Each nonexpansive local isometry of a metric continuum into itself is an isometry onto itself. 4. Each local isometry of a convex metric continuum into itself is an isometry onto itself.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 643-647
  • MSC: Primary 54E40
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0660621-7
  • MathSciNet review: 660621