TITLE: A class of sets which is closed under the formation of finite intersections necessarily contain the universal set.
QUESTION [0 upvotes]: How does a class of sets (may not be empty) which is closed under the formation of finite intersections necessarily contain the universal set? I came across this in G.F. Simmons (page-12).
This doubt might be very silly, but  I do not see how it follows. Can anyone please explain?

REPLY [2 votes]: To say that a family $\mathcal X$ of subsets of some universal set $U$ is closed under finite intersections means that, for any finite subfamily $\mathcal F\subseteq\mathcal U$, the intersection $\bigcap\mathcal F$ of all the sets in $\mathcal F$ is also in $\mathcal X$. In particular, since the empty family $\varnothing$ is always a finite subfamily of $\mathcal X$ (regardless of whether $\mathcal X$ itself is empty or not), and since $\bigcap\varnothing=U$, we get $U\in\mathcal X$.