Closed under intersections I read this definition: "A collection C of subsets of E is said to be closed under intersections if A ∩ B belongs to C whenever A and B belong to C."
How could the intersection of ANY A and B belonging to C ever NOT belong to C?? Whats the point of this definition?

Let $E=\{1, 2, 3\}$ and suppose the collection $C$ of subsets was $C=\{\{1, 2\}, \{2, 3\}\}$. Then this collection of two subsets is not closed under intersection, since $\{1, 2\}\cap\{2, 3\}=\{2\}$, which is not in $C$.
