TITLE: Isometry and equivalence
QUESTION [6 upvotes]: Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces on the same set $X$. Is there any relation between $d_1$ and $d_2$ being equivalent and $(X,d_1)$ and $(X,d_2)$ being isometric? If not, can anyone give examples where $d_1$ and $d_2$ are equivalent but $(X,d_1)$ and $(X,d_2)$ are not isometric; and where $d_1$ and $d_2$ are not equivalent, but $(X,d_1)$ and $(X,d_2)$ are isometric?

REPLY [2 votes]: If $(X,d_1),(X,d_2)$ are isometric metric spaces with identity map then they're equvalent,but otherwise,it's not true in generally,For example $X=\mathbb{R}-0$,$d_2$ is Euclidean metric and $d_1(x,y)=|\frac{1}{x}-\frac{1}{y}|$ now $f:(X,d_1)\to (X,d_2) $s,t $x\mapsto\frac{1}{x}$ is isometry but these are not equivalent because {$n$}$_{n\in\mathbb{N}}$ is Cauchy in $(X,d_1)$.