Definition:Topologically Equivalent Metric Spaces

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $f: A_1 \to A_2$ be a bijection such that:
 * $f$ is continuous from $M_1$ to $M_2$
 * $f^{-1}$ is continuous from $M_2$ to $M_1$.

Then:
 * $f$ is a topological equivalence
 * $M_1$ and $M_2$ are topologically equivalent.

Also known as
A topological equivalence is also known as a homeomorphism.

Two topologically equivalent metric spaces can be described as homeomorphic.

The definition of homeomorphism between topological spaces is fully compatible with this.

Also see

 * Metric Induces Topology
 * Equivalence of Definitions of Continuity on Metric Spaces


 * Definition:Homeomorphism (Topological Spaces)