Wallman compactification

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

is closed under finite intersections, and is maximal among those families that have these properties.

For every closed subset F of X, the class ΦF of points of ωX containing F is closed in ωX.

The topology of ωX is generated by these closed classes.