Proposition 4

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 $X$ into $Y_i$. Let $\mathscr{S}$ be the set of subsets of $X$ of the form $\overline{f}_i^{-1}(U_i)$ ($i \in I$, $U_i$ open in $Y_i$), and let $\mathfrak{B}$ be the set of finite intersections of sets of $\mathscr{S}$. Then $\mathfrak{B}$ is a base of a topology $\mathfrak{T}$ on $X$ which is the initial topological structure on $X$ for the family $(f_i)$ (Set Theory, Chapter IV, § 2, no. 3) and in particular is the coarsest topology on $X$ for which the mappings $f_i$ are continuous. More precisely, if $g$ is a mapping of a topological space $Z$ into $X$, then $g$ is continuous at a point $z \in Z$ (X carrying the topology $\mathfrak{T}$) if and only if each of the functions $f_i \circ g$ is continuous at $z$.

Let $\mathscr{D}$ be the set of all unions of sets belonging to $\mathfrak{B}$; clearly $\mathscr{D}$ satisfies axiom $(O_1)$ since formation of unions is associative; and $\mathscr{D}$ satisfies axiom $(O_{II})$ by reason of the definition of $\mathfrak{B}$ and the fact that finite intersection is distributive over arbitrary union [Set Theory, R § 4, formula (37)]. The set $\mathscr{D}$ is therefore the set of open subsets of $X$ for a topology $\mathfrak{T}$ of which $\mathfrak{B}$ is a base. We shall prove the last assertion of the proposition, which implies the others by reason of the general properties of initial structures (Set Theory, Chapter IV, § 2, no. 3, criterion CST 9). In the first place, the definition of $\mathscr{S}$ shows that the $f_i$ are continuous on $X$ (no. 1, Theorem 1); hence, if $g$ is continuous at $z$, so are the mappings $f_i \circ g$ (no. 1, Proposition 2). Conversely, suppose that all the mappings $f_i \circ g$ are continuous at $z$, and let $V$ be a neighbourhood of $g(z)$ in $X$; by definition, there is a finite subset $J$ of $I$, and for each $i \in J$ an open subset $U_i$ of $Y_i$ such that $V$ contains the set $\bigcap_{i \in J} \overline{f}_i^{-1}(U_i)$ and $g(z)$ belongs to this set. It follows that

and the hypothesis implies that each of the sets $\overline{g}^{-1}(\overline{f}_i^{-1}(U_i))$ is a neighbourhood of $z$ in $Z$; hence $\overline{g}^{-1}(V)$ is also a neighbourhood of $z$ in $Z$. This completes the proof.

Let $\mathfrak{B}_i$ be a base of the topology of $Y_i (i \in I)$; let $\mathscr{S}'$ denote the set of subsets of $X$ of the form $\overline{f}_i^{-1}(U_i)$ for $i \in I$ and $U_i \in \mathfrak{B}_i$ for each $i \in I$; if $\mathfrak{B}'$ is the set of finite intersections of sets of $\mathscr{S}'$, it is evident that $\mathfrak{B}'$ is a base of the topology $\mathfrak{T}$.

The general properties of initial structures (Set Theory, Chapter IV, § 2, no. 3, criterion CST 10) imply in particular the following transitivity property (the direct proof of which is quite straightforward):