Let $T_j = (X_j, \mathcal{T}_j)$ be a D1106: Topological space for each $j \in J$.
Let $f_j : X_j \to Y$ be a D18: Map for each $j \in J$.
The pushforward topology on $Y$ with respect to $f = \{ f_j \}_{j \in J}$ and $T = \{ T_j \}_{j \in J}$ is the D11: Set \begin{equation} \{ E \subseteq Y \mid \forall \, j \in J : f^{-1}_j(E) \in \mathcal{T}_j \} \end{equation}