Definition:Homeomorphism/Topological Spaces/Definition 2
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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.
$f$ is a homeomorphism if and only if:
- $\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$
That is, $f$ is a homeomorphism if and only if:
- for all subsets $U$ of $S_\alpha$, $U$ is open in $T_\alpha$ if and only if $f \sqbrk U$ is open in $T_\beta$.
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
- Results about homeomorphisms in the context of topological spaces can be found here.
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets: Definition $2$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms: Definition $3.6.1$