Regular language

A language is regular exactly when it is accepted by a finite automaton, or equivalently has finitely many Myhill-Nerode classes.

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.

For a fixed symbol $a$ and a regular language $L$, inserting exactly one $a$ at an arbitrary position preserves regularity. Make two copies of a recognizing finite-state automaton, one before and one after the insertion. Preserve its ordinary automaton transitions in each copy, and add a transition consuming $a$ from each before-state to its matching after-state. Accept only in the after-copy. Every accepting run makes exactly one of the additional moves.