basis of a final topology

1. Proposition

Let \(X\) be a set, \((X_i,\mathcal{T}_i)\) be a class of topological spaces and \(f_i: X_i \rightarrow X\) be a class of set-theoretic maps Then for the final topology

\begin{align*} \{U \subseteq X \vert \forall f_i : f_i^{-1}[U] \in \mathcal{T}_i\} \end{align*}