Definition:Homeomorphism/Topological Spaces/Definition 1

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 both $f$ and $f^{-1}$ are continuous.

Also see

 * Equivalence of Definitions of Homeomorphic Topological Spaces