Let $T_X = (X, \mathcal{T}_X)$ and $T_Y = (Y, \mathcal{T}_Y)$ each be a D1106: Topological space.
A D18: Map $f : X \to Y$ is continuous at $x_0 \in X$ with respect to $T_X$ and $T_Y$ if and only if \begin{equation} \forall \, U \in \mathcal{T}_X \left( x_0 \in U \quad \implies \quad \exists \, V \in \mathcal{T}_Y : U \subseteq f^{-1}(V) \right) \end{equation}