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 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.

gollark: The relevant parts of those are overridden by PotatOS privacy policy clause 4.2.γ, however.
gollark: That only works for certain HTTP headers, not all of them.
gollark: Oh, we had to disable that in most space-time because of people duplicating apioids.
gollark: I bet your proof relied on ridiculous axioms like the "law of the excluded middle" and "peano arithmetic".
gollark: I even had a truth cuboid validate it.

See also

References

  • Aleksandrov, P.S. (2001) [1994], "Wallman_compactification", Encyclopedia of Mathematics, EMS Press
  • Wallman, Henry (1938), Lattices and topological spaces, 39, pp. 112–126, JSTOR 1968717
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.