Definition:Homeomorphism/Topological Spaces

Definition
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.

Definition 4
$T$ and $T'$ are said to be homeomorphic.

The symbol $T \sim T'$ is often seen.

Equivalence of Definitions
See Equivalent Definitions for Homeomorphism for a proof that the definitions are equivalent.

Also known as
Also known as:
 * a topological equivalence, usually used when the spaces in question are metric spaces
 * an isomorphism.

Caution
Not to be confused with homomorphism.

Also see

 * Inverse of Homeomorphism is Homeomorphism