Metric characterization of connectedness for topological spaces

TL;DR

This paper introduces a unified metric framework for various notions of connectedness in topological and uniform spaces, bridging classical concepts and extending to a hierarchy of connectedness types.

Contribution

It provides a novel metric formalism that unifies connectedness, path connectedness, and uniform connectedness, and extends to a hierarchy of connectedness notions.

Findings

Unified metric formalism for connectedness concepts

Shows equivalence of connectedness and uniform connectedness in compact metric spaces

Extends connectedness notions to a hierarchy of concepts

Abstract

Connectedness, path connectedness, and uniform connectedness are well-known concepts. In the traditional presentation of these concepts there is a substantial difference between connectedness and the other two notions, namely connectedness is defined as the absence of disconnectedness, while path connectedness and uniform connectedness are defined in terms of connecting paths and connecting chains, respectively. In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. Connectedness in topological spaces can also be defined in terms of chains governed by open coverings in a manner that is more reminiscent of path connectedness. We present a unifying metric formalism for connectedness, which encompasses both connectedness of topological spaces and uniform connectedness of uniform spaces, and which further extends to a hierarchy of notions of connectedness.