Infinite intersection of regular languages

Regular languages are closed under finite intersection, but not countably infinite intersection. Indeed, for any formal language $K\subseteq {\Sigma }^{\ast }$ over a finite alphabet, let $L_i$ contain every word of length at least $i$, together with the members of $K$ of length less than $i$. Each $L_i$ is a regular language, since its exceptional short part is a finite language, and ${\bigcap }_{i\ge 1}{L}_i=K$. Thus every formal language, including one that is not a regular language, can arise in this way.