  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.
