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Proceedings of the American Mathematical Society

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Shape triviality and metric contractions

Author: W. Holsztyński
Journal: Proc. Amer. Math. Soc. 69 (1978), 199-200
MSC: Primary 54F43
MathSciNet review: 482618
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Abstract: Let (X, d) be a nonempty compact metric space such that for every $ \varepsilon > 0$ there exists a map $ f:X \to X$ satisfying

(i) $ d(x,f(x)) < \varepsilon $ for every $ x \in X$, and

(ii) $ d(f(x),f(y)) < d(x,y)$ for every $ x,y \in X$.

Then, as proved in this paper, the shape of X is trivial. This improves an earlier result of K. Borsuk [1], who proved that, under the same assumptions, X is acyclic.

References [Enhancements On Off] (What's this?)

  • [1] K. Borsuk, Concerning a problem due to Sam B. Nadler, Jr., Colloq. Math. 35 (1976), 63-65. MR 0431130 (55:4132)
  • [2] S. B. Nadler, Jr., Some problems concerning stability of fixed points, Colloq. Math. 27 (1973), 263-268. MR 0328897 (48:7239)

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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

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