# Wallman compactification

In mathematics, the **Wallman compactification**, generally called **Wallman–Shanin compactification** is a compactification of T_{1} topological spaces that was constructed by Wallman (1938).

## Definition

The points of the Wallman compactification ω*X* of a space *X* are the maximal proper filters in the poset of closed subsets of *X*. Explicitly, a point of ω*X* is a family of closed nonempty subsets of *X* such that 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 ω*X* containing *F* is closed in ω*X*. The topology of ω*X* is generated by these closed classes.

## Special cases

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

## See also

## References

- Aleksandrov, P.S. (2001) [1994], "Wallman_compactification", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Wallman, Henry (1938),
*Lattices and topological spaces*,**39**, pp. 112–126, JSTOR 1968717