\begin{definition}[Definition:Homeomorphism/Topological Spaces/Definition 2]
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''' {{iff}}:
:$\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$
That is, $f$ is a '''homeomorphism''' {{iff}}:
:for all subsets $U$ of $S_\alpha$, $U$ is open in $T_\alpha$ {{iff}} $f \sqbrk U$ is open in $T_\beta$.
\end{definition}