Definition:Homeomorphism/Topological Spaces
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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
- Results about homeomorphisms in the context of topological spaces can be found here.