# Notions of equivalent metrics

Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and $d'$, in increasing order of strength:

1. $i$ is a homeomorphism, i.e. $d$ and $d'$ induce the same topology on $X$.

2. $i$ and $i^{-1}$ are uniformly continuous.

3. $i$ is bilipschitz, i.e. $C_1 d' \le d \le C_2 d'$.

I would like to know what terms are used for these notions.

In particular, Mathworld says that the term "equivalent" refers to sense 1. This seems counterintuitive since, for instance, sense 1 does not preserve completeness. Munkres's General Topology uses "metrically equivalent" for sense 2. Does this agree with people's experience of standard usage?

Edit: I will point out that 3 implies 2 implies 1 (since Lipschitz implies uniformly continuous implies continuous) but converses are false. Let $X = \mathbb{R}$, let $d_1(x,y) = |x-y|$, $d_2(x,y) = |x-y| \wedge 1$, $d_3(x,y) = |\phi(x)-\phi(y)|$, where $\phi : \mathbb{R}\to (0,1)$ is your favorite homeomorphism. In sense 1 all three are equivalent, in sense 2 $d_1 \sim d_2 \not\sim d_3$, and in sense 3 all are inequivalent. In particular note that $d_1, d_2$ are complete but $d_3$ is not.

Edit: Further confusing the matter is the fact that if $X$ is a vector space and $d,d'$ are induced by norms $||\cdot||, ||\cdot||'$, then all three senses coincide, and sense 3 is usually taken as the definition of "equivalent norms."

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–  Qiaochu Yuan Sep 23 '10 at 18:14
Dave L. Renfro mentioned in a comment to a related question his essay on Lipschitz, uniformly, and topologically equivalent metrics. –  Martin Sleziak Nov 6 '11 at 8:14
There's a notion of "uniform space" that makes the idea of "uniform continuity" (and related things) make sense in general; number two, uniform equivalence, is saying that the metrics induce the same uniform structure (just as number 1 is saying they induce the same topology). –  Harry Altman Nov 6 '11 at 9:06

I don't know if there is a standard, but I can provide two sample points.

Burago, Burago, and Ivanov in "A course in metric geometry", p. 9 call definition (3) Lipschitz equivalent.

Dugundji in "Topology" calls those metrics satisfying (1) equivalent.

Burago et al. is available online.

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I can't seem to make a link work. –  yasmar Sep 23 '10 at 18:21

The first one is simply the definition of topological equivalence - it verbatim extends to general topological spaces (not necessarily metric spaces). I don't have a name for the second one. The third, which is used often, is called "Lipschitz equivalence".

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I would call (1) topologically equivalent or though as yasmar mentions some just use equivalent.

I would call (3) strongly equivalent but in numerical analysis we often use just equivalent (as in your norm equivalent example)

I'm fairly confident that (1) iff (2) and certain that (3) implies (1).

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Thanks! But 1 does not imply 2, see my edit. –  Nate Eldredge Sep 23 '10 at 18:42
@Nate: Thanks for the edit! Great counterexample! –  alext87 Sep 23 '10 at 18:55