Definition 2

Let $(X_l)_{l\in I}$ be a family of sets whose index set $I$ is not empty. The set $\mathscr{E}_x((\forall l)((l\in I)\Longrightarrow(x\in X_l)))$, that is to say, the set of all $x$ which belong to every set of the family $(X_l)_{l\in I}$, is called the intersection of the family and is denoted by

If $I=\varnothing$, the relation $(\forall l)((l\in I)\Longrightarrow(x\in X_l))$ is not collectivizing in $x$; for it is a true relation and there exists no set $Y$ such that $x\in Y$ is a true relation, because $Y$ would then be the set of all objects (cf. §1, no. 7, Remark).

If $(X_l)_{l\in I}$ is a family of subsets of a set $E$ and if $I\neq\varnothing$, the relation “$x\in E$ and $(\forall l)((l\in I)\Longrightarrow(x\in X_l))$” is equivalent to

consequently it is collectivizing in $x$, and the set of all $x$ which satisfy this relation is equal to $\displaystyle\bigcap_{l\in I}X_l$. If $I=\varnothing$, the relation “$x\in E$ and $(\forall l)((l\in I)\Longrightarrow(x\in X_l))$” is equivalent to $x\in E$; it is therefore collectivizing in $x$, and the set of all $x$ which satisfy this relation is $E$. Hence we may state the following definition :