Definition:Homeomorphism

This page is about Homeomorphism in the context of topology. For other uses, see Isomorphism.

Not to be confused with Definition:Homomorphism.

Definition

Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.

Let $f: T_\alpha \to T_\beta$ be a bijection.

$f$ is a homeomorphism if and only if both $f$ and $f^{-1}$ are continuous.

The same definition applies to 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.

A homeomorphism is also known as:

a topological equivalence, usually used when the spaces in question are metric spaces
an isomorphism, usually used when the spaces in question are manifolds.

Not to be confused with homomorphism.