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On rigid subsets of some manifolds. (English) Zbl 0706.54026
K. Borsuk proved that if every homeomorphism preserving length of arcs, between open connected subsets of a Euclidean space, is an isometry [Glasnik Mat. 16(36), 307-311 (1981; Zbl 0494.54027)]. The present paper generalizes the result to certain finite-dimensional manifolds and shows that it is not true for the Hilbert space.
Reviewer: M.Hušek

##### MSC:
 54E40 Special maps on metric spaces 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010)
##### Keywords:
homeomorphism preserving length of arcs; isometry
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