Nearly metacompact space

Definition
A topological space is said to be nearly metacompact if given any open cover, we can find a dense subset of the topological space, and a refinement of the open cover, that is point-finite on the dense subset.

Formalisms
Open $$\to$$ Open, point-finite on a dense subset

Stronger properties

 * Compact space
 * Paracompact space
 * Metacompact space