Shape triviality and metric contractions
Abstract: Let (X, d) be a nonempty compact metric space such that for every there exists a map satisfying
(i) for every , and
(ii) for every .
Then, as proved in this paper, the shape of X is trivial. This improves an earlier result of K. Borsuk , who proved that, under the same assumptions, X is acyclic.
-  Karol Borsuk, Concerning a problem due to Sam B. Nadler, Jr.: “Some problems concerning stability of fixed points” (Colloq. Math. 27 (1973), 263–268, 332), Colloq. Math. 35 (1976), no. 1, 63–65. MR 0431130, https://doi.org/10.4064/cm-35-1-63-65
-  Sam B. Nadler Jr., Some problems concerning stability of fixed points, Colloq. Math. 27 (1973), 263–268, 332. MR 0328897, https://doi.org/10.4064/cm-27-2-263-268
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Keywords: Shape theory, trivial shape, metric contractions approximating the identity map, acyclicity, fixed point
Article copyright: © Copyright 1978 American Mathematical Society