FAC Spaces

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

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.

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problems in well-structured transition systems, remains decidable if one relaxes “well-quasi-ordered” by “FAC” (unpublished, as far as I know).  A poset is FAC 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.

One of the keys to that discovery is the following theorem.

Thm (Kabil and Pouzet) [2, Lemma 5.3].  A poset is FAC if and only if every downwards-closed subset is a finite union of ideals.

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

Whence my question:

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?

Maybe that is known, but I don’t know.  [Corrigendum, October 30th, 2017: that problem is now solved.]

A few things I know:

  • The FAC spaces that are also Alexandroff spaces are exactly the Alexandroff spaces of a FAC poset.  (That should come as no surprise.)
  • Every topological space X such that every closed subspace C has a dense Noetherian subspace D, is FAC.
    This is easy: D is closed in itself and Noetherian, hence is a finite union I1I2 ∪ ··· ∪ In of irreducible closed subsets of D.  Let cl(A) denote the closure of A in X, for any subset A of X.  By density, C=cl(D)=cl(I1) ∪ cl(I2) ∪ ··· ∪ cl(In), and each cl(Ik) is irreducible closed in X [1, Lemma 8.4.10].
    This argument is similar to one in Kabil and Pouzet’s proof.  Here is how they show that every downwards-closed subset C in a FAC space must be a finite union of ideals.  They notice that ideals and irreducible closed subsets are the same thing in that setting.  Next, by an observation due to Hausdorff, C has a cofinal well-founded subset D.  Then D, being well-founded and FAC, is well-quasi-ordered.  It follows that D is a finite union of ideals I1I2 ∪ ··· ∪ In (ideals in D, not C). Write ↓A for downwards-closure of A in X.  We now check that, by cofinality, C=↓D=↓I1 ∪ ↓I2 ∪ ··· ∪ ↓In, and each ↓Ik is an ideal in X.
  • I conjecture that the converse is wrong.  Let me ask that in a neutral form: are the FAC spaces exactly those such that every closed subspace has a dense Noetherian subspace?
  • Since the property of being a FAC space only depends on its lattice of closed sets, a topological space X is FAC if and only if its sobrification SX is FAC.
  • The sober FAC spaces are exactly the spaces where every closed set is finitary.  A finitary closed set is a set of the form ↓{x1, x2, ···,  xn}, where ↓ denotes downward closure with respect to the specialization ordering ≤.  This is obvious, considering that, in a sober space, every irreducible subset is the closure of a (unique) point.  In particular, the topology of a sober FAC space is the upper topology of ↓, since that is the coarsest topology where every finitary closed subset is closed.
  1. Jean Goubault-Larrecq. Non-Hausdorff Topology and Domain Theory — Selected Topics in Point-Set Topology. New Mathematical Monographs 22. Cambridge University Press, 2013.
  2. M. Kabil and M. Pouzet. Une extension d’un théorème de P. Julien sur les âges de mots. Informatique théorique et applications, 26(5):449–482, 1992.

Jean Goubault-Larrecq(February 23rd, 2017)jgl-2011