Definition:Homeomorphism/Topological Spaces

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.

Definition 1

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

Definition 2

$f$ is a homeomorphism if and only if:

$\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$

Definition 3

$f$ is a homeomorphism if and only if $f$ is both an open mapping and a continuous mapping.

Definition 4

$f$ 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 known as

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.

Also see

• Equivalence of Definitions of Homeomorphisms between Topological Spaces

• Inverse of Homeomorphism is Homeomorphism

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