Topology as faithful communication through relations

TL;DR

This paper introduces a new interpretation of topological concepts as faithful communication through relations, showing that open and closed sets can be characterized by this communication, and that continuous functions are precisely those that can be faithfully communicated.

Contribution

It provides a novel perspective on topology by interpreting topological notions as faithful communication via relations, linking continuity with faithful communication.

Findings

Open and closed sets are exactly those that can be faithfully communicated.

A relation or function can be faithfully communicated if and only if it is continuous.

The framework offers a new interpretation of topological concepts through relations.

Abstract

Basic pairs and their morphisms are the most elementary framework in which standard topological notions can be defined. We present here a new interpretation of topological concepts as those which can be communicated faithfully between the two sides of basic pairs. In particular, we prove that the subsets which can be communicated faithfully (in the suitable way) are exactly open subsets and closed subsets. We also prove that a relation (and in particular a function) between two sets of points can be communicated faithfully if and only if it is continuous.