\begin{definition}[Definition:Continuous Mapping (Topology)/Point/Neighborhoods]
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 neighborhood $N$ of $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.
\end{definition}