Let $d_1$ and $d_2$ be two metrics on a space $M$ such that the metric spaces $(M, d_1)$ and $(M, d_2)$ are homeomorphic to each other. I know that if the identity map is continuous, then metrics are equivalent.
However, I am not able to go beyond this. In other words, I can neither prove nor able to produce a counterexample to the statement that
homeomorphism between the spaces implies the metrics are equivalent.
My definition of equivalence of metrics is there exists $\alpha,\beta$ such that for every $x,y\in M$, $$\alpha d_1(x,y)\leq d_2(x,y)\leq \beta d_1(x,y).$$ As the spaces are homeomorphic, an open set in one space is open in other. But this may not imply that a $\epsilon$-ball in one space is a $\delta$-ball in other.
Can someone help me the clarifying this?
Does the situation become different in the case of normed linear spaces?