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