Definition 1

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets (resp. a family of subsets of a set E). The set $\mathscr{E}_x((\exists \iota)(\iota \in \mathrm{I}$ and $x \in \mathrm{X}_\iota))$, that is to say, the set of all $x$ which belong to at least one set of the family $(X_l)_{l\in I}$, is called the union of the family, and is denoted by

[^1]

If $(X_l)_{l\in I}$ is a family of subsets of a set $E$, then its union is a subset of $E$; notice that it does not depend on $E$, nor on the target $\mathfrak{G}$ of the mapping $l\to X_l$.

It is clear that if $I=\varnothing$, we have $\displaystyle\bigcup_{l\in I}X_l=\varnothing$, because the relation $(\exists l)(l\in I\text{ and }x\in X_l)$ is then false.

¶ Suppose now that $I\neq\varnothing$. If $\alpha$ is an element of $I$, the relation

implies $x\in X_\alpha$ and therefore, by virtue of C52 (§1, no. 6), this relation is collectivizing in $x$.