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
,
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