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).

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 Φ_F 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

• Lattice (order)
• Pointless topology

References

• Aleksandrov, P.S. (2001) [1994], "Wallman_compactification", Encyclopedia of Mathematics, EMS Press
• Wallman, Henry (1938), "Lattices and topological spaces", Annals of Mathematics, 39 (1): 112–126, doi:10.2307/1968717, JSTOR 1968717