Definition:Homeomorphism/Topological Spaces/Definition 2

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.

$f$ is a homeomorphism iff:
 * $\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \left({U}\right) \in \tau_\beta$

That is, $f$ is a homeomorphism iff
 * for all subsets $U$ of $S_\alpha$, $U$ is open in $T_\alpha$ iff $f \left({U}\right)$ is open in $T_\beta$.

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 $T_\beta$.

Also see

 * Equivalence of Definitions of Homeomorphism