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

The principle of pointfree continuity

In the setting of constructive pointfree topology, we introduce a notion of continuous operation between pointfree topologies and the corresponding principle of pointfree continuity. An operation between points of pointfree topologies is continuous if it is induced by a relation between the bases of the topologies; this gives a rigorous condition for Brouwer's continuity principle to hold. The principle of pointfree continuity for pointfree topologies $\mathcal{S}$ and $\mathcal{T}$ says that any relation which induces a continuous operation between points is a morphism from $\mathcal{S}$ to $\mathcal{T}$. The principle holds under the assumption of bi-spatiality of $\mathcal{S}$. When $\mathcal{S}$ is the formal Baire space or the formal unit interval and $\mathcal{T}$ is the formal topology of natural numbers, the principle is equivalent to spatiality of the formal Baire space and formal unit interval, respectively. Some of the well-known connections between spatiality, bar induction, and compactness of the unit interval are recast in terms of our principle of continuity. We adopt the Minimalist Foundation as our constructive foundation, and positive topology as the notion of pointfree topology. This allows us to distinguish ideal objects from constructive ones, and in particular, to interpret choice sequences as points of the formal Baire space

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

北海道における知的障がい者の就労支援に関する一考察

知的障がい者の就労について、北海道及び北海道教育委員会が進めている障が いのある人の就労支援の充実に向けた取組の状況を概観することに加えて、北海道内 の特別支援学校在籍者の約８割を占めている知的障がい特別支援学校の現状や就労支 援の取組について整理した。北海道において障がいある人の就労に大きな役割を果た してきた職親会の設立の経緯やなよろ地方職親会の障がい者雇用の状況やジョブコー チ養成研修の成果をまとめた。以上のことを踏まえて、知的障がい者の就労支援やキ ャリア教育の在り方について考察する

Reale e ideale in matematica

Quaderni di Acme 12

Subdirectly irreducible modal algebras and initial frames

initial frame

Two applications of dynamic constructivism: Brouwer's continuity principle and choice sequences in formal topology

In dynamic constructivism the origin of concepts is seen to be a dialectical process between two requirements: convenience of abstractions and faithfulness to reality. The essence of constructivism is then shifted and becomes awareness of the level of abstraction and its uses, rather than a static self-limitation to certain principles. This is perfectly consonant with a minimalist foundation of mathematics, which in particular is based on two different levels, one for computational (intensional) and one for geometrical (extensional) aspects of mathematics. After a short general introduction, dynamic constructivism is illustrated by two specific applications, which exploit formal topology over a minimalist foundation. My (silent) claim is that this attitude could be consonant to Brouwer's spirit (if not letter). Mathematically, it brings = some new light on two controversial topics of intuitionism, since Brouwer's time. I will show under which assumptions Brouwer's principle, saying that all functions on the real numbers are continuous, can be proved and generalized. And I will argue for a rigourous and simple definition of choice sequences