TRANSITIVITY IN POINT-FREE TOPOLOGY

From intuitionistic to point-free topology

Brouwer’s pioneering results in topology, e.g. invariance of dimension, were developed within a classical framework of mathematics. Some years later he explained that “in his topological work he tried to use only methods which he expected could be made constructive” [11, p. XIV]. It seems that very little of algebraic topology, homotopy and homology, has actually been developed constructively i...

The Canonicity of Point-free Topology

Properties of topological spaces are not absolute, as they may depend on the ambient mathematical universe. For instance, the answers to some questions in classical point-set topology hinge on non-standard set-theoretic axioms, while in intuitionistic mathematics even such basic properties as the Heine-Borel compactness of the closed interval depend on the ambient constructive variety. We show ...

Topology and Measure in Logics for Point-Free Space

Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. But philosophers have for a long time doubted whether there are any point-sized regions of space. On an alternative, region-based approach to space, extended regions together with the relations of parthood and contact are taken as primitive; points are represented as mathematical ab...

A Point-Free Relation-Algebraic Approach to General Topology

On the T1 axiom and other separation properties in constructive point-free and point-set topology

In this note a T1 formal space (T1 set-generated locale) is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of T ∗ 1 formal space, and prove that the class of points of a weakly set-presentable T ∗ 1 formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulat...