{"id":374,"date":"2014-06-24T19:10:49","date_gmt":"2014-06-24T17:10:49","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=374"},"modified":"2022-11-19T15:31:57","modified_gmt":"2022-11-19T14:31:57","slug":"filters-v-wallman-compactifications","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=374","title":{"rendered":"Filters V: Wallman compactifications"},"content":{"rendered":"

The Wallman compactification.<\/strong><\/p>\n

In Filters IV<\/a>, we have shown that we could realize the Stone-\u010cech compactification \u00dfX<\/em> of a discrete space X<\/em> as its space of ultrafilters of subsets U<\/strong>X<\/em>.\u00a0 The topology on the latter had the sets U<\/em>\u266f<\/sup> as a basis, where U<\/em>\u266f<\/sup> = {F<\/em> in U<\/strong>X <\/em>| U<\/em> in F<\/em>}.<\/p>\n

In 1938, Henry Wallman gave a more general construction [1], which applies to any T1<\/sub> space X<\/em>, and produces the Stone-\u010cech compactification \u00dfX<\/em> of X<\/em> (up to isomorphism) whenever X is T4<\/sub>.\u00a0 I will describe it in detail.\u00a0 Many extensions were given since thence, notably by Orrin Frink in 1964 [2], but I will not say anything about them.<\/p>\n

Last time, I’ve also warned you that this post would be technical.\u00a0 I guess you’ll agree it holds to its promises.\u00a0 I’ve tried to postpone the hard, Stone-duality based material, as much as I could.\u00a0 Still, this post is going to be packed.<\/p>\n

Instead of considering ultrafilters of subsets, that is, maximal non-trivial filters of subsets, Wallman considers maximal non-trivial filters in the complete lattice H<\/strong>X<\/em> of closed subsets of X<\/em>.\u00a0 (The H<\/strong> is for Hoare: H<\/strong>X<\/em> is the so-called Hoare powerdomain over X, up to a small detail.\u00a0 I will talk about it another time.)\u00a0 Let us write \u03c9<\/strong>X<\/em> for the subset of maximal non-trivial filters of closed subsets of X<\/em>.\u00a0 I’ll say Wallman filter<\/em> (over X) instead of “maximal non-trivial filter of closed subsets of X”, which starts to be lengthy.<\/p>\n

We do as in Filters IV, replacing subsets by closed subsets, ultrafilters of subsets by Wallman filters, and U<\/strong>X<\/em> by \u03c9<\/strong>X<\/em>.<\/strong><\/p>\n

In U<\/strong>X<\/em>, we had defined A<\/em>\u266f<\/sup> as the set of ultrafilters of subsets that contained A<\/em>, and these were the opens (and also the closed sets) of U<\/strong>X<\/em>.\u00a0 We would like to adapt this to \u03c9<\/strong>X<\/em>.\u00a0 Reasoning with opens is awkward, here, however, since Wallman filters are filters of closed subsets.\u00a0 So we shall define C<\/em>\u266f<\/sup> only when C<\/em> is a closed subset of X, and declare these sets C<\/em>\u266f<\/sup> to be closed<\/em>.\u00a0 In other words, the complements<\/em> of these sets form a subbasis for a topology on \u03c9<\/strong>X<\/em>.\u00a0 We shall call it the Wallman topology<\/em>.<\/p>\n

Theorem.<\/strong> \u03c9<\/strong>X<\/em> is a compact T1<\/sub> space.<\/p>\n

Proof.<\/em>\u00a0 As for U<\/strong>X<\/em>, we do this with filters.
\n<\/em><\/p>\n