Definition:Topologically Equivalent Metrics

Definition
Let $M_1 = \left({A, d_1}\right)$ and $M_2 = \left({A, d_2}\right)$ be metric spaces on the same underlying set $A$.

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


 * $U \subseteq A$ is $d_1$-open $\iff$ $U \subseteq A$ is $d_2$-open.

Also see

 * Equivalence of Definitions of Topologically Equivalent Metrics


 * Topological Equivalence is Equivalence Relation


 * Metrics are Topologically Equivalent iff Continuity Preserved


 * Definition:Homeomorphism