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
If such a homeomorphism exists, then $T_\alpha$ and $T_\beta$ are said to be homeomorphic.

The symbolism $T_\alpha \sim T_\beta$ is often seen to denote that $T_\alpha$ is homeomorphic to $T_\beta$.

Equivalence of Definitions
See Equivalence of Definitions of Homeomorphic Topological Spaces 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.

Also see

 * Inverse of Homeomorphism is Homeomorphism