When does infinite intersection preserve a closed property?

Question

There are two statements well known in Math and Computer Science:

1. Intersection of infinite number of regular languages is not regular.
2. Intersection of infinite number of convex sets is convex.

Notice that in both cases, a finite intersection preserves the corresponding property:

1. Intersection of finite number of regular languages is regular.
2. Intersection of finite number of convex sets is convex.

I.e, both regular languages and convex sets are closed under intersection.

In most literature the fact about infinite intersection of convex sets is treated as obvious and not requiring any proofs. Before I learnt the things about regular languages, I would conclude the same about them. What is so special about regular languages which leads us to a different conclusion? Why is it everybody relies on the intuition in the case with convex sets and why it does not work for regular languages?

Answer 1

What is so special about regular languages which leads us to a different conclusion?

Well, the set of all regular languages contains the singletons and is closed under finite union (by definition) and under complement (a consequence of Kleene's theorem). It is also known that there are some non-regular languages. Now take any set of subsets of an infinite set $E$ containing the singletons, closed under finite union and complement and not equal to $\mathcal{P} (E)$. Then this set of subsets will not be closed under infinite intersection.