Definition:Topologically Equivalent Metrics/Definition 2

Definition

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

$d_1$ and $d_2$ are topologically equivalent if and only if:

$U \subseteq A$ is $d_1$-open if and only if $U \subseteq A$ is $d_2$-open.

Also see

• Equivalence of Definitions of Topologically Equivalent Metrics

• Results about topologically equivalent metrics can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Proposition $2.4.2$