Example 1

Let $\mathfrak{S}$ be a non-empty subset of the set $\mathfrak{P}(E)$ of subsets of a set E. If $\mathfrak{S}$ has a least (resp. greatest) element A with respect to the inclusion relation, then A is the intersection (resp. union) of the sets of $\mathfrak{S}$. Conversely, if the intersection (resp. union) of the sets of $\mathfrak{S}$ belongs to $\mathfrak{S}$, then it is the least (resp. greatest) element of $\mathfrak{S}$.