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arXiv · 2607.18502

From Regional Topology to Point-Class Topology in Tarski's Geometry of Solids

Patrick Barlatier·Richard Dapoigny
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Abstract

Tarski's geometry of solids reconstructs point-like entities from concentric families of spherical regions rather than taking points as primitive. We formalize this reconstruction in Coq within a nominal mereological framework inspired by Le\'sniewski, and study how the regional geometry of solids induces a topology on reconstructed point-classes without adding new point individuals. Three of Tarski's postulates concerning solids and interior points, P2--P4, are derived as theorems. We refine Tarski's interior-point notion and define a regional interior operator satisfying the four Kuratowski interior axioms, together with a boundary operation and an encoding of RCC8 relations. We then pass to the setoid of ball representatives modulo same_center. Each ball generates a stable basic point-plural GBasicPointSet(Q), and these plurals form a basis for a metatheoretic topology on reconstructed point-classes. This topology is Hausdorff under Tarski's separation axiom Three_points and non-discrete under the local richness hypothesis BallCenterBundle. Finally, geometric neighbourhoods yield a closure operator GClosurePoint satisfying the four Kuratowski closure axioms and respecting extensional point-set equality. The formalization thus verifies the passage from a regional topology of solids to a Hausdorff topology and neighbourhood closure on reconstructed point-classes.

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BibTeXRIS

Patrick Barlatier, Richard Dapoigny. 2026-07-20. From Regional Topology to Point-Class Topology in Tarski's Geometry of Solids. https://arxiv.org/abs/2607.18502

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