Abstract
Let (X,dX) and (Y,dY) 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 dX=f∘dY∘(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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