38 research outputs found

### Overlap Algebras as Almost Discrete Locales

Boolean locales are almost discrete. In fact, spatial Boolean locales are the
same thing as discrete spaces. This does not make sense intuitionistically,
since (non-trivial) discrete locales fail to be Boolean. We show that Sambin's
"overlap algebras" have good enough features to be called "almost discrete
locales".
Keywords. Strongly dense sublocales, almost discrete spaces, overlap
algebras, constructive topology

### $\sigma$-locales in Formal Topology

A $\sigma$-frame is a poset with countable joins and finite meets in which
binary meets distribute over countable joins. The aim of this paper is to show
that $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of
Formal Topology, that is, intuitionistic and predicative point-free topology.
Every $\sigma$-frame $L$ is the lattice of Lindel\"of elements (those for which
each of their covers admits a countable subcover) of a formal topology of a
specific kind which, in its turn, is a presentation of the free frame over $L$.
We then give a constructive characterization of the smallest (strongly) dense
$\sigma$-sublocale of a given $\sigma$-locale, thus providing a
``$\sigma$-version'' of a Boolean locale. Our development depends on the axiom
of countable choice.Comment: Paper presented at the conference Continuity, Computability,
Constructivity - From Logic to Algorithms (CCC 2017), Nancy, France, June
26-30 201

### Positivity relations on a locale

This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)

### Embedding locales and formal topologies into positive topologies

A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure
of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a re\ufb02ective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales

### Overlap Algebras: a Constructive Look at Complete Boolean Algebras

The notion of a complete Boolean algebra, although completely legitimate in
constructive mathematics, fails to capture some natural structures such as the
lattice of subsets of a given set. Sambin's notion of an overlap algebra,
although classically equivalent to that of a complete Boolean algebra, has
powersets and other natural structures as instances. In this paper we study the
category of overlap algebras as an extension of the category of sets and
relations, and we establish some basic facts about mono-epi-isomorphisms and
(co)limits; here a morphism is a symmetrizable function (with classical logic
this is just a function which preserves joins). Then we specialize to the case
of morphisms which preserve also finite meets: classically, this is the usual
category of complete Boolean algebras. Finally, we connect overlap algebras
with locales, and their morphisms with open maps between locales, thus
obtaining constructive versions of some results about Boolean locales.Comment: Postproceedings of CCC2018: Continuity, Computability,
Constructivity. Faro, Portugal, 24-28 Sep 201

### Constructive version of Boolean algebra

The notion of overlap algebra introduced by G. Sambin provides a constructive
version of complete Boolean algebra. Here we first show some properties
concerning overlap algebras: we prove that the notion of overlap morphism
corresponds classically to that of map preserving arbitrary joins; we provide a
description of atomic set-based overlap algebras in the language of formal
topology, thus giving a predicative characterization of discrete locales; we
show that the power-collection of a set is the free overlap algebra
join-generated from the set. Then, we generalize the concept of overlap algebra
and overlap morphism in various ways to provide constructive versions of the
category of Boolean algebras with maps preserving arbitrary existing joins.Comment: 22 page

### A constructive Galois connection between closure and interior

We construct a Galois connection between closure and interior operators on a
given set. All arguments are intuitionistically valid. Our construction is an
intuitionistic version of the classical correspondence between closure and
interior operators via complement.Comment: This is a revised version. Content is reorganized so to separate
clearly what requires an impredicative proof from what can be proven also
predicatively. Moreover, some results are given in a more general form and
some counterexamples are adde

### Reducibility, a constructive dual of spatiality

An intuitionistic analysis of the relationship between pointfree and pointwise topology brings new notions to light that are hidden from a classical viewpoint. In this paper, we study one of these, namely the notion of reducibility for a pointfree topology, which is classically equivalent to spatiality. We study its basic properties and we relate it to spatiality and to other concepts in constructive topology. We also analyse some notable examples. For instance, reducibility for the pointfree Cantor space amounts to a strong version of Weak K\uf6nig\u2019s Lemma

### Overlap Algebras as Almost Discrete Locales

Boolean locales are "almost discrete", in the sense that a spatial Boolean
locale is just a discrete locale (that is, it corresponds to the frame of open
subsets of a discrete space, namely the powerset of a set). This basic fact,
however, cannot be proven constructively, that is, over intuitionistic logic,
as it requires the full law of excluded middle (LEM). In fact, discrete locales
are never Boolean constructively, except for the trivial locale. So, what is an
almost discrete locale constructively? Our claim is that Sambin's overlap
algebras have good enough features to deserve to be called that. Namely, they
include the class of discrete locales, they arise as smallest strongly dense
sublocales (of overt locales), and hence they coincide with the Boolean locales
if LEM holds