Weak similarities of metric and semimetric spaces

Oleksiy Dovgoshey; Evgeniy Petrov

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arXiv · 1209.1820

Weak similarities of metric and semimetric spaces

Oleksiy Dovgoshey·Evgeniy Petrov

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Abstract

Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X \to Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) \to D(X) such that dX = f \circ dY \circ F. It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F : X \to Y is an isometry if X and Y are ultrametric and compact with D(X) = D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.

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BibTeX RIS

Oleksiy Dovgoshey, Evgeniy Petrov. 2012-09-09. Weak similarities of metric and semimetric spaces. https://arxiv.org/abs/1209.1820

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