Abstract
Let (X,d X ) and (Y,d Y ) be semimetric spaces with distance sets D(X) and D(Y), respectively. A map** 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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Dovgoshey, O., Petrov, E. Weak similarities of metric and semimetric spaces. Acta Math Hung 141, 301–319 (2013). https://doi.org/10.1007/s10474-013-0358-0
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DOI: https://doi.org/10.1007/s10474-013-0358-0