Equivalence of metrics

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In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or equivalent.

In the following, M will denote a non-empty set and d₁ and d₂ will denote two metrics on M.

[edit] Topological equivalence

The two metrics d₁ and d₂ are said to be topologically equivalent if they generate the same topology on M. There are many equivalent ways of expressing this condition:

• a subset is d₁-open if and only if it is d₂-open;
• the open balls "nest": for any point and any radius r > 0, there exist radii r',r'' > 0 such that

and

• the identity function is both (d₁,d₂)-continuous and (d₂,d₁)-continuous.

The following is a sufficient but not necessary condition for topological equivalence:

• for each , there exist constants c₁,c₂ > 0 such that, for every point ,