Weak similarities of metric and semimetric spaces |
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Authors: | Oleksiy Dovgoshey Evgeniy Petrov |
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Affiliation: | 1. Institute of Applied Mathematics and Mechanics of NASU, R. Luxemburg str. 74, Donetsk, 83114, Ukraine
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Abstract: | Let (X,d X ) and (Y,d Y ) be semimetric spaces with distance sets D(X) and D(Y), respectively. A mapping F:?X→Y is a weak similarity if it is surjective and there exists a strictly increasing f:?D(Y)→D(X) such that d X =f°d Y °(F?F). It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F:?X→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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