Dimension on Discrete Spaces

TL;DR

This paper introduces combinatorial models for continuous spaces, defining dimension on discrete spaces through axioms, and explores their applications to various geometric and topological structures, including spacetime.

Contribution

It develops a framework for defining dimension on discrete spaces and constructs models of classical geometric objects with topological singularities.

Findings

Discrete models of Euclidean spaces and spheres

Construction methods for new discrete spaces

Analysis of topology in spacetime models

Abstract

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S.