Topological Exterior

The exterior of a subset $S\subseteq X$ , denoted $\mathrm{ext}(S)$ of a Topological Space $X$ is the largest open set that is disjoint from $S$ . More specifically, it is the union

$\par\mathrm{ext}(S)=\bigcup\left\{\;U\in\mathcal{O}({X})\right\}$

The exterior of a subset is the topological analog of a Complement of a Subset, and thus carries all of the latters Constructive pitfalls.

Properties

Clasically, the exterior of $S$ is the Complement of its Interior, though this may not hold Constructively.
The exterior of any open set is a Regular Open Set.
The exterior operator induces a Monad on the Frame of opens; this is analogous to the Double Negation Monad.