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Proceedings of the American Mathematical Society

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

 
 

 

Local isometries of compact metric spaces


Author: Aleksander Całka
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
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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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Keywords: Locally nonexpansive mapping, nonexpansive mapping, local isometry, isometry, decomposition of the space, convex space
Article copyright: © Copyright 1982 American Mathematical Society