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 $$f'$$ are continuous.

Then $$f$$ is a homeomorphism.

Otherwise known as a homeomorphism.