Understanding equivalent metric spaces 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
 A: Two definitions on equivalence are not the same. Definition 2 implies definition 1, but not vice versa.
For instance, Let $X = (0, 1]$ and $d_1(x, y) = |x - y|$ and $d_2(x, y) = |\frac 1 x - \frac 1 y|$. Then $d_1$ is equivalent to $d_2$ under Def-1, but not under Def-2. Indeed, one can see $X$ is complete under $d_1$, but not $d_2$.
A: 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: 


*

*$\mathrm{id}$ is a homeomorphism 

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

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

*$\mathrm{id}$ is a quasisymmetric homeomorphism

*$\mathrm{id}$ is an isometry

*$\mathrm{id}$ 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 $3\implies 6$.
