Statures and Sobrification Ranks of Noetherian Spaces

Abstract

There is a rich theory of maximal order types of well-partial-orders (wpos), pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is Noetherian in its Alexandroff topology, and there are more; this prompts us to investigate an analogue of that theory in the wider context of Noetherian spaces.The notion of maximal order type does not seem to have a direct analogue in Noetherian spaces per se, but the equivalent notion of stature, investigated by Blass and Gurevich (2008) does: we define the stature ||X|| of a Noetherian space X as the ordinal rank of its poset of proper closed subsets. We obtain formulas for statures of sums, of products, of the space of words on a space X, of the space of finite multisets on X, in particular. They confirm previously known formulas on wpos, and extend them to Noetherian spaces.The proofs are, by necessity, rather different from their wpo counterparts, and rely on explicit characterizations of the sobrifications of the corresponding spaces, as obtained by Finkel and the first author (2020).We also give formulas for the statures of some natural Noetherian spaces that do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces, powersets, and spaces of words on X with the socalled prefix topology.Finally, because our proofs require it, and also because of its independent interest, we give formulas for the ordinal ranks of the sobrifications of each of those spaces, which we call their dimension.

Infinitary Noetherian Constructions II. Transfinite Words and the Regular Subword Topology

“…We write ||F || for the rank of F in the lattice of closed subsets of a Noetherian space X. For any Noetherian space F , ||F || is the stature of F [8], generalizing the notion of the same name on wqos [1].…”

“…For every ordinal α, let α • be α+1 if α = δ+n for some critical ordinal δ and some n ∈ N, and α otherwise. Then α < ω α • , and α → α • is strictly monotonic [8,Lemmata 12.3 and 12.4]. For every proper bound γ, we define [γ] as the rank of γ in the poset of all proper bounds less than or equal to γ.…”