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I have studied following definitions of equivalent metric spaces.

Two metrics on a set $X$ are said to be equivalent if and only if they induce the same topology on $X$.

1: Two metrices $d_1$ and $d_2$ in metric space $X$ are equivalent if $d_1(x_n,x_0)\rightarrow 0 $ iff $d_2(x_n,x_0)\rightarrow 0 $.

2: We say that d1 and d2 are equivalent iff there exist positive constants $c$ and $C$ such that $c d_1(x, y)\leq d_2(x, y)\leq Cd_1(x, y)$ for all $x, y \in X$.

My questions are as follows:

Is there any other definition of equivalent metrics? I need a proof of how these conditions are equivalent?

Is there any connection between homeomorphism and equivalence of metric spaces?

What are the common properties shared by equivalent metric spaces?

I am very much confused with this. Quite often I found myself struggling with what definition should I apply to show the equivalence of given metric spaces. I need help to clear my doubts.

Thanks a lot for helping me

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The conditions 1. and 2. are not equivalent. 1. expresses that $d_1$ and $d_2$ have the same closed sets. 2. expresses that $d_1$ and $d_2$ have the same Cauchy sequences (and more). Property 2. implies property 1. The metrics $d_1(x,y) = |x-y|$ and $d_2(x,y) = \min\{1,|x-y|\}$ on $X = \mathbb{R}$ are equivalent in first sense but not in the second sense. A third sense (in between 1 and 2) is mentioned in this thread. – t.b. May 28 '12 at 12:19
up vote 4 down vote accepted

The short answer to "Is there any connection between homeomorphism and equivalence of metric spaces?" is yes. The long answer: any reasonable notion of equivalence of two metrics $d_1$ and $d_2$ can be formulated in terms of the identity map $\mathrm{id}\colon (X,d_1)\to (X,d_2)$. As soon as we distinguish a class of "nice" maps (the class should be a group under composition), we get a notion of equivalence. Some examples, previously mentioned and not:

  1. $\mathrm{id}$ is a homeomorphism

  2. $\mathrm{id}$ is a uniform homeomorphism (i.e., uniformly continuous with a uniformly continuous inverse)

  3. $\mathrm{id}$ is a bilipschitz homeomorphism (i.e., Lipschitz with a Lipschitz inverse)

  4. $\mathrm{id}$ is a quasisymmetric homeomorphism

  5. $\mathrm{id}$ is an isometry

  6. $\mathrm{id}$ is a quasi-isometry

... The list is not exhaustive.

The corresponding notions of equivalence are related by $5\implies 3\implies 2\implies 1$, also $3\implies 4\implies 1$, and $3\implies 6$.

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