Monotonically normal space

Definition with symbols
A topological space $$X$$ is termed monotonically normal if it is a T1 space (i.e., all points are closed) and there exists an operator $$G$$ from ordered pairs of disjoint closed sets to open sets, such that:


 * 1) For any disjoint closed subsets $$A,B$$, $$G(A,B)$$ contains $$A$$ and its closure is disjoint from $$B$$
 * 2) If $$A \subseteq A'$$ and $$B' \subseteq B$$ with all four sets being closed, $$A$$ disjoint from $$B$$, and $$A'$$ disjoint from $$B'$$, we have:

$$G(A,B) \subseteq G(A',B')$$

This is the monotonicity condition. Such an operator $$G$$ is termed a monotone normality operator.

Incomparable properties

 * Perfectly normal space

Metaproperties
Any subspace of a monotonically normal space is monotonically normal.