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 closed with respect to $T_X$ and $T_Y$ if and only if \begin{equation} \forall \, E \in \mathcal{P}(X) \, (X \setminus E \in \mathcal{T}_X \quad \implies \quad Y \setminus f(E) \in \mathcal{T}_Y) \end{equation}