In mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain properties. This notion generalizes equivalence of norms to arbitrary metric spaces.

will denote a non-empty set.

Topological equivalence

are said to be topologically equivalent if they generate the same topology. The adverb topologically is often dropped. There are multiple ways of expressing this condition:

• -open if and only if-open;

• the open balls

• the identity function is continuous with continuous inverse; that is, it is a homeomorphism.

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

• there exists a strictly increasing, continuous, and subadditive.

Strong equivalence

on are strongly or bilipschitz equivalent (sometimes also uniformly equivalent,

.

are topologically equivalent, but not strongly equivalent. In fact, this interval is bounded under one of these metrics but not the other. On the other hand, strong equivalences always take bounded sets to bounded sets.

Equivalence of metrics from perspective of categories

is a category whose objects are all metric spaces and morphisms are some continuous maps (i.e. a wide subcategory of Top.

. Other examples include to take morphisms to be all similarities, or all uniformly continuous maps, obtaining uniform equivalence. Uniformly equivalent metrics in this sense do not are uniformly but not strongly equivalent.

Relation with equivalence of norms

When are those induced by norms For linear operators between normed vector spaces, Lipschitz continuity is equivalent to continuity—an operator satisfying either of these conditions is called bounded. are simply said to be equivalent.

In finite dimensional vector spaces, all metrics induced by a norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are equivalent.

Properties preserved by equivalence

• The continuity of a function is preserved if either the domain or range is remetrized by an equivalent metric, but uniform continuity is preserved only by strongly equivalent metrics.

• The differentiability a subset of a normed space, is preserved if either the domain or range is renormed by an equivalent norm.

• A metric that is strongly equivalent to a complete metric is Cauchy but not convergent. (It is not Cauchy in the induced metric.)

See also

• Equivalence (measure theory)

Notes

References