Closure Property
Definition. A class of languages is closed under an operation if applying that operation to languages in the class always yields a language in the class. A statement asserting this is a closure property.
The regular languages are closed under union, concatenation, and star, which follows immediately from regular expressions, and under complement, by swapping the accepting states of a DFA, and intersection, by the product construction. Closure properties shorten non-regularity proofs: if a language would let you build a known non-regular language using operations that preserve regularity, it cannot be regular, which is how Lecture 4 avoids applying the pumping lemma more than once.