A Whiteheadian-type description of Euclidean spaces, spheres, tori and Tychonoff cubes

TL;DR

This paper constructs algebraic dual objects for Euclidean spaces, spheres, tori, and Tychonoff cubes using dualities from Whitehead's region-based space theory, providing a new algebraic characterization of these topological spaces.

Contribution

It introduces a direct algebraic construction of dual objects for key topological spaces based on existing dualities, advancing Whitehead's philosophical ideas into precise mathematical form.

Findings

Algebraic dual objects fully characterize Euclidean spaces and related topological spaces.

The construction is direct, without relying on the topological spaces themselves.

Provides a new algebraic perspective on classical topological spaces.

Abstract

In the beginning of the 20th century, A. N. Whitehead and T. de Laguna proposed a new theory of space, known as {\em region-based theory of space}. They did not present their ideas in a detailed mathematical form. In 1997, P. Roeper has shown that the locally compact Hausdorff spaces correspond bijectively (up to homeomorphism and isomorphism) to some algebraical objects which represent correctly Whitehead's ideas of {\em region} and {\em contact relation}, generalizing in this way a previous analogous result of de Vries concerning compact Hausdorff spaces (note that even a duality for the category of compact Hausdorff spaces and continuous maps was constructed by de Vries). Recently, a duality for the category of locally compact Hausdorff spaces and continuous maps, based on Roeper's results, was obtained by G. Dimov (it extends de Vries' duality mentioned above). In this paper, using the dualities obtained by de Vries and Dimov, we construct directly (i.e. without the help of the corresponding topological spaces) the dual objects of Euclidean spaces, spheres, tori and Tychonoff cubes; these algebraical objects completely characterize the mentioned topological spaces. Thus, a mathematical realization of the original philosophical ideas of Whitehead and de Laguna about Euclidean spaces is obtained.