{"id":1141,"date":"2017-02-23T18:51:17","date_gmt":"2017-02-23T17:51:17","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1141"},"modified":"2022-05-16T15:43:04","modified_gmt":"2022-05-16T13:43:04","slug":"fac-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1141","title":{"rendered":"FAC Spaces"},"content":{"rendered":"

A Noetherian space is a space in which every open is compact, see [1], Section 9.7 for example.
\nNoetherian spaces have the very nice property that every closed subset is a finite<\/em> union of irreducible closed subsets, and that generalizes the theorem (attributed to Erd\u00f6s and Tarski) that, in a well-quasi-ordered set, every downwards-closed subset is a finite union of ideals.<\/p>\n

That property has been used by me and my colleagues repeatedly over the last few years, and is fundamental in the study of so-called well-structured transition systems, which are transition systems in which the state space is well-quasi-ordered, and in which the transition relation is monotonic.<\/p>\n

In 2016, Alain Finkel discovered that coverability, one of the fundamental decidable problems in well-structured transition systems, remains decidable if one relaxes “well-quasi-ordered” by “FAC” (unpublished, as far as I know).\u00a0 A poset is FAC<\/em> if and only if it has the finite antichain property, namely, if and only if all its antichains (sets of pairwise incomparable elements) are finite.<\/p>\n

One of the keys to that discovery is the following theorem.<\/p>\n

Thm<\/strong> (Kabil and Pouzet) [2, Lemma 5.3].\u00a0 A poset is FAC if and only if every downwards-closed subset is a finite union of ideals.<\/p>\n

Hence one has a full characterization of those posets in which downwards-closed subsets are finite union of ideals.\u00a0 Well-quasi-orders have this property, but the spaces that have this property, more generally, are all the FAC posets.\u00a0 For example, the poset of integers, Z<\/strong>, is FAC, but not well-quasi-ordered, since it is not well-founded.\u00a0 In general, every totally ordered space is FAC.<\/p>\n

Whence my question:<\/p>\n

Is there a similar characterization of those topological spaces (call them FAC spaces) in which every closed subset is a finite union of irreducible closed subsets?<\/p><\/blockquote>\n

Maybe that is known, but I don’t know.\u00a0 [Corrigendum, October 30th, 2017: that problem is now solved<\/a>.]<\/p>\n

A few things I know:<\/p>\n