regulaarsetel

Regulaarsetel is a term used in Estonian topology to denote a class of subsets of a topological space that satisfy a regularity condition with respect to the interior and closure operations. In practice, there are two closely related notions commonly described under this umbrella: regular open sets and regular closed sets.

Regular open sets: a set U is regular open if U = int(cl(U)). Regular closed sets: a set F is regular closed if F = cl(int(F)). In many texts, regulaarsetel is used as a generic label, and authors specify which variant they mean.

Examples help illustrate the idea. In the real line with the standard topology, any open interval (a,b) is regular open, since int(cl((a,b))) = int([a,b]) = (a,b). A closed interval [a,b] is regular closed, since cl(int([a,b])) = cl((a,b)) = [a,b]. Some sets, such as (a,b], are not regular open nor regular closed.

Usage and context: The term regulaarsetel appears mainly in Estonian-language texts on topology and geometry; in English-language literature these concepts are typically referred to as regular open and regular closed sets. The regulaarsetel notion emphasizes the shared idea that a set is determined by the interaction of interior and closure, even though the two standard variants are defined separately as regular open and regular closed.