Weak similarities of metric and semimetric spaces

TL;DR

This paper investigates weak similarities in semimetric spaces, showing they are equivalent to usual similarities in geodesic spaces and are isometries in ultrametric compact spaces with equal distance sets.

Contribution

It establishes conditions under which weak similarities coincide with traditional similarities, isometries, or homeomorphisms in various classes of semimetric spaces.

Findings

Weak similarities in geodesic spaces are actual similarities.

Weak similarities are isometries in ultrametric compact spaces with equal distance sets.

Conditions for weak similarities to be homeomorphisms or uniform equivalences are identified.

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.