Definition:Homeomorphism/Topological Spaces/Definition 4

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 $f$ is both a closed mapping and a continuous mapping.

Terminology

Let a homeomorphism exist between $T_\alpha$ and $T_\beta$.

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 $T_\beta$.

Also see

• Equivalence of Definitions of Homeomorphisms between Topological Spaces

• Results about homeomorphisms in the context of topological spaces can be found here.