The Canonicity of Point-free Topology

Properties of topological spaces are not absolute, as they may depend on the ambient mathematical universe. For instance, the answers to some questions in classical point-set topology hinge on non-standard set-theoretic axioms, while in intuitionistic mathematics even such basic properties as the Heine-Borel compactness of the closed interval depend on the ambient constructive variety. We show that, in contrast, there is a robust and natural formulation of point-free topology: the category of countably presented topologies (or locales, or formal spaces) is determined up to equivalence by the number-theoretic functions of the ambient framework. As most varieties of constructive mathematics agree that such functions are the Turing computable ones, they all have the same notion of countably-presented point-free spaces. (This talk presents joint work with Alex Simpson, University of Ljubljana.) A spatiality-like property for pointfree topologies with a positivity relation Francesco Ciraulo University of Padova It is an unchangeable fact of history that pointfree topology appeared after pointwise one. For this reason the former is “condemned” to compare itself with the latter and, in particular, to consider the case of spatial topologies as a notable case. I will talk about one of the constructive manifestations of the classical notion of spatiality. This has recently been isolated by Sambin [4], who gave it the name of reducibility. It was then studied in [1], of which I will present some results. Here are two examples: reducibility for the pointfree version of Cantor space amounts to the Weak König’s Lemma; reducibility for the pointfree Baire space states that every element of a (not necessarily decidable) spread belongs to a choice sequence contained in that spread [3]. In general, reducibility is a point existence property. Reducibility was born in the more general context of formal topologies with a positivity relation, also called positive topologies, where it emerged for structural motivations and reasons of symmetry. In [2] we managed to give a “localic” description of the category of positive topologies; in view of that work, reducibility can be now understood as statement about the weakly closed sublocales of the given locale. Classically, reducibility is equivalent to spatiality, at least in the particular case of locales; and assuming either spatiality implies reducibility or the converse yields the full law of excluded middle. In the more general case of positive topologies, instead, even if classical logic is assumed, reducibility remains distinct from, actually weaker than, spatiality. This makes the notion of reducibility potentially interesting also to a classical mathematician.