On Tarski's fixed point theorem

A concept of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive and predicative version of Tarski's fixed point theorem is obtained.Comment: Proc. Amer. Math. Soc., to appea

Constructive metrisability in point-free topology

AbstractThe notion of elementary diameter is introduced to provide, in the context of Locale Theory, a constructive notion of metrisability. Besides foundational aspects, elementary diameters allow to express metrisability in locales more simply with respect to the existing (non-constructive) approach based on diameters. By relying on the presentation of Locale Theory provided by formal topology, the notions to be presented may be conceived as phrased within (Martin-Löf) Type Theory. A type-theoretic version of Urysohn metrisation theorem is thus obtained. As an application, a set (data type) of indexes for the points of locally compact metrisable formal spaces is shown to exist

On the existence of Stone-Čech compactification

Introduction. In 1937 E. Čech and M.H. Stone independently introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 18]. In the introduction of [8] the non-constructive character of this result is so described: “it must be emphasized that β(S) [the Stone-Čech compactification of S] may b

Exact approximations to Stone-\u10cech compactification.

Many fundamental results as the theorems of Tychonoff and Hahn-Banach are equivalent, in classical ZF, to some form of choice principle. A well-known advantage of replacing topological spaces with locales, or formal spaces, is that topos-valid (choice-free) versions of these results become often provable. Generally speaking, this remains true even if one replaces topoi with constructive (intuitionistic and predicative) settings such as Type Theory or Constructive Set Theory. This paper deals with the rather extreme case of Stone-\u10cech compactification f. The formal spaces/ locales X for which f(X) exists constructively are characterized. This involves discussing the 'size' of hom-sets in the category Loc of locales/formal spaces (constructively Loc is not locally small). In particular, the full subcategory of locally compact regular locales is proved to be locally small. It is also shown how Stone-\u10cech compactification can itself be used to prove that certain hom-sets are small

On the collection of points of a formal space

none1nononeCuri, GiovanniCuri, Giovann

On the existence of Stone-\u10cech compactification.

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