Regular Open Set

A subset of a Topological Space $X$ is a regular open set if it is equal to the Exterior of its Exterior.

Properties

• In Classical Mathematics, the collection of all regular open sets of a space $X$ forms a Complete Boolean Algebra:

  • The join $U\wedge V$ is $\mathrm{Int}(\overline{U\cup V})$
  • The meet $U\vee V$ is $U\cap V$
  • The complement $\lnot U$ is $\mathrm{Int}(X\backslash U)$

  In Constructive Mathematics, we need to be a bit more careful; instead, the regular open sets form an Overlap Algebra.