Definition:Topologically Equivalent Metrics

This page is about topological equivalence of metrics on the same underlying set. For other uses, see Definition:Topological Equivalence.

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 $A$.

Definition 1

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

For all metric spaces $\left({B, d}\right)$ and $\left({C, d'}\right)$:
For all mappings $f: B \to A$ and $g: A \to C$:
$(1): \quad f$ is $\left({d, d_1}\right)$-continuous if and only if $f$ is $\left({d, d_2}\right)$-continuous
$(2): \quad g$ is $\left({d_1, d'}\right)$-continuous if and only if $g$ is $\left({d_2, d'}\right)$-continuous.

Definition 2

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

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