Definition:Homeomorphism

Topological Spaces
Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

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

Metric Spaces
The same definition applies to metric spaces:

Manifolds
The same definition applies to manifolds:

Also known as
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.

Caution
Not to be confused with homomorphism.

Also see

 * Definition:Topological Equivalence


 * Inverse of Homeomorphism is Homeomorphism