Choice Sequences vs. Formal Topology

It is a fact of classical mathematics that the adjunction O a Pt : Loc→ Sp is not an equivalence which, however, boils down to an equivalence between sober spaces and spatial locales. However, classically, most of the usual spaces are sober as e.g. all Hausdorff spaces and all algebraic domains. But there are notable examples of non-spatial locales which do not have any points as e.g. the measure algebra (Borel subsets of R identified when their symmetric difference has measure 0) or the regular open subsets of R. Locales, i.e. point free topology, have their origin in Grothendieck’s topos theory. The (2-)category Top of Grothendieck toposes and geometric morphisms hosts the full reflective sub-(2-)category of localic Grothendieck toposes equivalent to Loc. In a classical framework ordinary mathematics usually does not require the consideration of pointless spaces, i.e. locales which haven’t got enough points. But, of course, they are important for constructing Boolean (or Heyting) valued models of, say, set theory and related systems. E.g. for refuting the Axiom of Choice one may consider the boolean valued model V (B) where B is the measure algebra, i.e. Borel subsets of R modulo the equivalence relation identifying Borel sets A and B iff their symmetric difference A∇B has Lebesgue measure 0. In any case many non-atomic complete boolean algebras haven’t got enough points (e.g. regular open subsets of R give rise to a complete Boolean algebra without any points). Constructively, however, many locales haven’t go enough points because their construction requires non-constructive means. For example in the effective topos Eff formal Cantor space hasn’t got enough points because the Kleene tree provides a family of basic open sets which do not cover formally but cover all recursive binary sequences. This applies even more so to formal Baire space which observation by Kleene was his motivation to consider function realizability which provides a model for Brouwerian intuitionism. Function realizability is based on the partial combinatory algebra K2 (2nd Kleene algebra) whose underlying set is the Baire space NN, i.e. arbitrary sequences of natural numbers including the non-effective ones). The applica-