\begin{definition}[Definition:Continuous Mapping (Topology)/Point/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 $S_2$.
Let $x \in S_1$.
The mapping $f$ is '''continuous at (the point) $x$''' (with respect to the topologies $\tau_1$ and $\tau_2$) {{iff}}:
:For every open set $U_2$ of $T_2$ such that $\map f x \in U_2$, there exists an open set $U_1$ of $T_1$ such that $x \in U_1$ and $f \sqbrk {U_1} \subseteq U_2$.
\end{definition}