Definition:Homeomorphism/Topological Spaces/Definition 2

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

That is, $f$ is a homeomorphism :
 * for all subsets $U$ of $S_\alpha$, $U$ is open in $T_\alpha$ $f \sqbrk U$ is open in $T_\beta$.

Also see

 * Equivalence of Definitions of Homeomorphic Topological Spaces