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?
— Mike
P.S.:
Does the situation become different in the case of normed linear spaces?
Best Answer
A nice example is the square root metric on $\mathbb R$.
Define $d(x,y)=\sqrt{|x-y|}$.
This is a metric on $\mathbb R$ and induces the Euclidean topology. But there exists no constants $\alpha,\beta>0$ such that $\alpha d(x,y)<|x-y|<\beta d(x,y)$.