Category:Definitions/Homeomorphisms (Metric Spaces)

This category contains definitions related to homeomorphisms in the context of metric spaces.
Related results can be found in Category:Homeomorphisms (Metric Spaces).

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 homeomorphism

$M_1$ and $M_2$ are homeomorphic.