Localic completion of uniform spaces

We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set X equipped with a family of generalised metrics on X, where a generalised metric on X is a map from the product of X to the upper reals satisfying zero self-distance law and triangle inequality. For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. This point-free structure induces a complete generalised uniform structure on the set of formal points of the localic completion that gives the standard completion of the original gus with Cauchy filters. We extend the localic completion to a full and faithful functor from the category of locally compact uniform spaces into that of overt locally compact completely regular formal topologies. Moreover, we give an elementary characterisation of the cover of the localic completion of a locally compact uniform space that simplifies the existing characterisation for metric spaces. These results generalise the corresponding results for metric spaces by Erik Palmgren. Furthermore, we show that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus. We work in Aczel's constructive set theory CZF with the Regular Extension Axiom. Some of our results also require Countable Choice.Comment: 39 page

Localic Metric spaces and the localic Gelfand duality

In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C*-algebras. In order to do so we develop a constructive theory of localic metric spaces and localic Banach spaces, we study the notion of localic completion of such objects and the behaviour of these constructions with respect to pull-back along geometric morphisms.Comment: 57 page

Uniform locales and their constructive aspects

Much work has been done on generalising results about uniform spaces to the pointfree context. However, this has almost exclusively been done using classical logic, whereas much of the utility of the pointfree approach lies in its constructive theory, which can be interpreted in many different toposes. Johnstone has advocated for the development of a constructive theory of uniform locales and wrote a short paper on the basic constructive theory via covering uniformities, but he never followed this up with a discussion of entourage uniformities or completions. We present a more extensive constructive development of uniform locales, including both entourage and covering approaches, their equivalence, completions and some applications to metric locales and localic algebra. Some aspects of our presentation might also be of interest even to classically minded pointfree topologists. These include the definition and manipulation of entourage uniformities using the internal logic of the geometric hyperdoctrine of open sublocales and the emphasis on pre-uniform locales. The latter leads to a description of the completion as the uniform reflection of the pre-uniform locale of Cauchy filters and a new result concerning the completion of pre-uniform localic rings, which can be used to easily lift addition and multiplication on $\mathbb{Q}$ to $\mathbb{R}$ (or $\mathbb{Q}_p$) in the pointfree setting.Comment: 27 pages, minor edit

Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters