Definition:Homeomorphism/Topological Spaces/Definition 1

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

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