Proposition 5

Let $X$ be a set, $(Z_i)_{i \in I}$ a family of topological spaces, $(J_\lambda)_{\lambda \in L}$ a partition of $I$ and $(Y_\lambda)_{\lambda \in L}$ a family of sets indexed by $L$. Also for each $\lambda \in L$ let $h_\lambda$ be a mapping of $X$ into $Y_\lambda$; for each $\lambda \in L$ and each $i \in J_\lambda$ let $g_{i\lambda}$ be a mapping of $Y_\lambda$ into $Z_i$, and put $f_i = g_{i\lambda} \circ h_\lambda$. If each $Y_\lambda$ carries the coarsest topology for which the mappings $g_{i\lambda} (i \in J_\lambda)$ are continuous, then the coarsest topology on $X$ for which the $f_i$ are continuous is the same as the coarsest topology for which the $h_\lambda$ are continuous.