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Plasticity in metric spaces. (English) Zbl 1083.54016
Recall that a mapping \(f\) from a metric space \((X,d)\) into itself is called non-contractive if \(d(f(x),f(y))\geq d(x,y)\) for all \(x,y\in X\). The authors call a metric space \((X,d)\) an EC-space if every noncontractive bijection from \(X\) onto itself is an isometry. A metric space that is not an EC-space is called an NEC-space. The following results are obtained. Theorem 1. Every totally bounded metric space is an EC-space. Theorem 2. The set \(\mathbb{Z}\) of integers with the usual metric is an EC-space. Theorem 3. \(\mathbb{R}\setminus\mathbb{Z}\) with the usual metric is an EC-space. Theorem 4. A convex subset of the Euclidean space \(\mathbb{R}^n\) is a hereditary EC-space if and only if it is bounded. Theorem 5. If \((X,d)\) is a connected, compact, metric space, then \(X\times\mathbb{Z}\) (endowed with the usual product metric) is an EC-space. Theorem 6. \([0,1)\times\mathbb{Z}\) is an NEC-space (although \([0,1]\times\mathbb{Z}\) is an EC-space by the preceding theorem). Theorem 7. If \(C\) is the Cantor set, then \(C\times \mathbb{Z}\) is an NEC-space. Theorem 8. Every unbounded metric space with at least one accumulation point contains an NEC-space. Theorem 9. If \(X\) is an NEC-space, then \(X\times Y\) is an NEC-space for any metric space \(Y\).

54E35 Metric spaces, metrizability
Full Text: DOI
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