Regulieropen

Regulieropen, commonly known in English as regular open, denotes a class of subsets of a topological space with a specific closure–interior relationship. In a topological space X, a subset U is regulieropen if U equals the interior of its closure: U = int(cl(U)). Such sets are always open, since int(cl(U)) is open.

Characterizations and relationships

A set U is regulieropen precisely when it can be written as the interior of a closed set (F = cl(U)). Equivalently, U is regular open if and only if its complement X \ U is a regular closed set, where a set C is regular closed when C = cl(int(C)). The operation reg(A) = int(cl(A)) maps any subset A of X to a regular open set; applying reg to a regular open set returns the same set.

Algebraic structure

The collection RO(X) of regular open sets in X forms a natural lattice under inclusion. The meet is given by intersection U ∧ V = U ∩ V, and the join by U ∨ V = int(cl(U ∪ V)). The bottom and top elements are ∅ and X, respectively. There is a dual correspondence between RO(X) and the family of regular closed sets: taking complements establishes a connection between the two classes.

Examples

In the real line with the usual topology, every open set is regular open, since the interior of the closure of an open set recovers the original set. More generally, in many common spaces, open sets that are unions of open regions without boundary pathologies are regular open; spaces with less regular topologies can have open sets that are not regular open.

Importance

Regular open sets and their algebraic structure provide a tool for studying the organization of open sets and their closures, and they appear in areas such as domain theory and the study of Boolean algebras associated with topologies.