arXiv subjects

Giovanni Sambin

Publications and source records attributed to Giovanni Sambin.

5 recordsLinked to original sources

arXiv2026

Positive Topology and Feasible Refinement: Forcing Matrices, Positivity, and Information

We develop a conceptual and operational account of Positive Topology starting from a basic relation between points or models and observable properties. From this relation, two complementary structures emerge. The first captures universal refinement and cover: what must hold across all relevant cases and how information can be systematically refined. The second captures positivity and witnessed existence: what can be positively realized and sustained without relying on classical complements. A central result shows that the underlying relation between points and observables can be reconstructed from either of these induced structures. We also clarify the distinction between point-based and pointfree formulations: when points are available, positivity can be derived from the underlying forcing relation, while in the formal pointfree setting positivity is taken as primitive and its compatibility with cover is imposed axiomatically. We develop two complementary interpretations of the framework. The first is information-theoretic, viewing cover as refinement of partial information and positivity as witnessed feasibility. The second is game-theoretic, viewing positivity as the ability of a witness or hypothesis to survive successive legitimate refinements. The final part of the paper is deliberately programmatic. We outline how resource constraints, verification costs, and finite budgets can be incorporated into the framework. This leads to resource-sensitive notions of forcing, cover, positivity, and refinement, and raises new questions about how these structures behave as available resources change. Examples from medical diagnosis, legal reasoning, and AI systems illustrate the potential relevance of the approach to grounded, explainable, and resource-aware inference.

cs.LO↗

arXiv2019

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.

cs.LO↗

arXiv2016

Topology as faithful communication through relations

Basic pairs and their morphisms are the most elementary framework in which standard topological notions can be defined. We present here a new interpretation of topological concepts as those which can be communicated faithfully between the two sides of basic pairs. In particular, we prove that the subsets which can be communicated faithfully (in the suitable way) are exactly open subsets and closed subsets. We also prove that a relation (and in particular a function) between two sets of points can be communicated faithfully if and only if it is continuous.

math.LO↗

arXiv2012

Convergence in Formal Topology: a unifying notion

Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous definition is a special case. This leads to a predicative presentation and inductive generation of locales (formal covers), commutative quantales (convergent covers) and suplattices (basic covers) in a uniform way. Thanks to our abstract treatment of convergence, we are able to specify categorically the precise sense according to which our inductively generated structures are free, thus refining Johnstone's coverage theorem. We also obtain a natural and predicative version of a fundamental result by Joyal and Tierney: convergent covers (commutative quantales) correspond to commutative co-semigroups over the category of basic covers (suplattices).

math.LO↗

arXiv2012

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.

math.LO↗