Wallman compactification 📖 Wikipedia

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 ${\displaystyle {\mathcal {F}}}$ 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.

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

• Lattice (order)
• Pointless topology

• 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