Metrics on Space are Topologically Equivalent iff Identity Mapping is Homemorphism

Theorem
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.

Let $I_A$ denote the identity mapping on $A$.

Then:
 * $d_1$ and $d_2$ are topologically equivalent


 * $I_A: M_1 \to M_2$ is a homeomorphism.
 * $I_A: M_1 \to M_2$ is a homeomorphism.