Espaces connectifs : repr\'esentations, feuilletages, ordres, diff\'eologies

TL;DR

This paper extends the theory of connectivity spaces by generalizing connectivity order, introduces connective foliations, and explores relations with diffeological spaces, providing new mathematical frameworks and characterizations.

Contribution

It generalizes connectivity order to all spaces, develops connective foliations, and studies functorial relations with diffeological spaces, advancing the mathematical understanding of connectivity structures.

Findings

Connectivity order now applies to all connectivity spaces

Introduction of connective foliations as a new concept

Characterization of diffeologisable connectivity spaces

Abstract

This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case is now generalised to all connectivity spaces, and so to connective foliations. Finally, we start the study of some functorial relations between connectivity and diffeological spaces, and we give a characterization of diffeologisable connectivity spaces.