Prove that two metrics $d$ and $\rho$ on a set $M$ are equivalent if and only if the identity map on M is a homeomorphism from $(M,d)$ to $(M,\rho)$.

Question

Prove that two metrics $d$ and $\rho$ on a set $M$ are equivalent if and only if the identity map on M is a homeomorphism from $(M,d)$ to $(M,\rho)$.

I have no problem with the forward part of the proof. But, I am unable to produce any valid reasons for the backward part i.e to Prove:

If the identity map on M is a homeomorphism from $(M,d)$ to $(M,\rho)$ then prove that $d$ and $\rho$ are equivalent.

Thanks!

Answer 1

The identity $\operatorname {id}:(M,d)\to (M,\rho)$ is continuous iff for any $U$ open in $\rho$, we have $\operatorname {id}^{-1}(U)=U$ open in $d$.

Now do the same for the inverse: $\operatorname {id}:(M,\rho)\to(M,d)$. It's continuous iff $U$ open in $d$ implies $U$ open in $\rho$.

The claim follows, since both the identity and its inverse are continuous, when the identity is a homeomorphism.

Answer 2

If you define the metrics to be equivalent iff they induce the same topology, then this is trivial: the identity between a set with two topologies is a homeomorphism iff the two topologies are identical.