Proposition 6

Let $X$ be a set, let $(Y_i)_{i \in I}$ be a family of topological spaces, and for each $i \in I$ let $f_i$ be a mapping of $Y_i$ into $X$. Let $\mathscr{D}$ be the set of subsets $U$ of $X$ such that $\overline{f}_i^{-1}(U)$ is open in $Y_i$ for each $i \in I$; then $\mathscr{D}$ is the set of open subsets of $X$ in a topology $\mathfrak{T}$ on $X$ which is the final structure on $X$ for the family $(f_i)$ (Set Theory, Chapter IV, § 2, no. 5), and in particular $\mathfrak{T}$ is the finest topology on $X$ for which the mappings $f_i$ are continuous. In other words, if $g$ is a mapping of $X$ into a topological space $Z$, then $g$ is continuous ($X$ carrying the topology $\mathfrak{T}$) if and only if each of the mappings $g \circ f_i$ is continuous.

It is immediately verified that $\mathscr{D}$ satisfies the axioms $(O_I)$ and $(O_{II})$. [Set Theory, R, § 4, formulae (34) and (46)]. We shall prove the last assertion of the proposition, which implies the other assertions by reason of the general properties of final structures (Set Theory, Chapter IV, § 2, no. 5, criterion CST 18). It is clear that the $f_i$ are continuous in the topology $\mathfrak{T}$, by the definition of $\mathscr{D}$ (no. 1, Theorem 1); hence if $g$ is continuous, so is each mapping $g \circ f_i$ (no. 1, Theorem 2). Conversely, suppose that each $g \circ f_i$ is continuous, and let $V$ be an open set in $Z$; by hypothesis, $\overline{f}_i^{-1}(\overline{g}^{-1}(V))$ is open in $Y_i$ for each $i \in I$; hence $\overline{g}^{-1}(V) \in \mathscr{D}$, and the proof is complete.