Lipschitz Equivalent Metrics are Topologically Equivalent

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 $d_1$ and $d_2$ be Lipschitz equivalent.

Then $d_1$ and $d_2$ are topologically equivalent.

Also see

 * Lipschitz Equivalent Metric Spaces are Homeomorphic