Wallman compactification

Definition

The points of the Wallman compactification ωX of a space X are the maximal proper filters in the poset of closed subsets of X. Explicitly, a point of ωX is a family F of closed nonempty subsets of X such that F is closed under finite intersections, and is maximal among those families that have these properties. For every closed subset F of X, the class Φ_F of points of ωX containing F is closed in ωX. The topology of ωX is generated by these closed classes.

Special cases

For normal spaces, the Wallman compactification is essentially the same as the Stone–Čech compactification.