Wallman compactification

In mathematics, the Wallman compactification, generally called Wallman–Shanin compactification is a compactification of T1 topological spaces that was constructed by.

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 $$\mathcal F$$ of closed nonempty subsets of X such that $$\mathcal F$$ is closed under finite intersections, and is maximal among those families that have these properties. For every closed subset F of X, the class ΦF 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.