Wallman compactification

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


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 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 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.

See also


  • 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
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