Equivalence of Metrics - Topological Equivalence

Topological Equivalence

The two metrics and are said to be topologically equivalent if they generate the same topology on . The adjective "topological" is often dropped. There are multiple ways of expressing this condition:

• a subset is -open if and only if it is -open;
• the open balls "nest": for any point and any radius, there exist radii such that

and

• the identity function is both -continuous and -continuous.

The following are sufficient but not necessary conditions for topological equivalence:

• there exists a strictly increasing, continuous, and subadditive such that .
• for each, there exist positive constants and such that, for every point ,