Definition 3

Let $(X_l)_{l\in I}$ be a family of subsets of a set $E$. The set

in other words, the set of all $x$ which belong to E and to each of the sets $X_\iota$, is called the intersection of the family and is denoted by $\bigcap\limits_{\iota \in I} X_\iota$.

Hence for a family $(X_\iota)_{\iota \in \emptyset}$ of subsets of E we have

But for a family $(X_\iota)_{\iota \in I}$ of subsets of E whose index set is not empty, the intersection $\bigcap\limits_{\iota \in I} X_\iota$ depends neither on E nor on the target of $\iota \rightarrow X_\iota$; and this justifies the use of the same notation in Definitions 2 and 3.