Geometric Spaces with No Points

TL;DR

This paper introduces models of set theory where certain 'geometric' spaces lack points, yet retain properties like distance functions and normable Riesz spaces, challenging traditional notions of geometry.

Contribution

It constructs set-theoretic models of geometric spaces without points, demonstrating that key geometric structures can exist without actual points.

Findings

Existence of point-free geometric models

Spaces with distance-like functions without points

Retention of Riesz space properties without points

Abstract

Some models of set theory are given which contain sets that have some of the important characteristics of being geometric, or spatial, yet do not have any points, in various ways. What's geometrical is that there are functions to these spaces defined on the ambient spaces which act much like distance functions, and they carry normable Riesz spaces which act like the Riesz spaces of real-valued functions. The first example has a family of sets, each one of which cannot be empty, but not in a uniform manner, so that it is false that all of them are inhabited. In the second, we define one fixed set which does not have any points, while retaining all of these geometrical properties.