Definition:Topologically Equivalent Metrics

Metric
Let $A$ be a set upon which there are two metrics imposed: $d_1$ and $d_2$.

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.

Alternative Definition
Let $A$ be a set upon which there are two metrics imposed: $d_1$ and $d_2$.

Let $\left({B, d}\right)$ and $\left({C, d\,'}\right)$ be any metric spaces.

Let $f: B \to A$ and $g: A \to C$ be any mappings such that:
 * $f$ is $\left({d, d_1}\right)$-continuous iff $f$ is $\left({d, d_2}\right)$-continuous;
 * $g$ is $\left({d_1, d\,'}\right)$-continuous iff $g$ is $\left({d_2, d\,'}\right)$-continuous.

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

Topological equivalence is clearly an equivalence relation.

Equivalence of Definitions
The above two definitions are equivalent.

Metric Spaces
Let $M$ and $M'$ be metric spaces.

Let $f: M \to M'$ be a bijection such that both $f$ and $f^{-1}$ are continuous.

Then $f$ is a topological equivalence.

Otherwise known as a homeomorphism.