Shape triviality and metric contractions

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.