# Wallman compactification

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In mathematics, the Wallman compactification, generally called Wallman–Shanin compactification is a compactification of T1 topological spaces that was constructed by Wallman (1938).

## 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 ${\displaystyle {\mathcal {F}}}$ of closed nonempty subsets of X such that ${\displaystyle {\mathcal {F}}}$ 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.