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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 $$Y$$.

##### MSC:
 5.4e+36 Metric spaces, metrizability
##### Keywords:
totally bounded metric space; EC-space; NEC-space
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##### References:
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