Definition:Topologically Equivalent Metric Spaces

Definition
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 or homeomorphism.

This definition also follows directly from:
 * The fact that a metric induces a topology
 * Equivalence of Metric Space Continuity Definitions.

Also known as
A homeomorphism in the context of metric spaces is usually referred to as topological equivalence.