\begin{definition}[Definition:Continuous Mapping (Topology)/Everywhere/Open Sets]
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.
The mapping $f$ is '''continuous on $S_1$''' {{iff}}:
:$U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$
where $f^{-1} \sqbrk U$ denotes the preimage of $U$ under $f$.
\end{definition}