Illustrating an error in "An equivalent condition for a uniform space to be coverable"

TL;DR

The paper identifies an error in Plaut's proof regarding the conditions under which a locally uniformly joinable uniform space is coverable, clarifying the relationship between these concepts.

Contribution

It highlights a specific mistake in Plaut's proof about the connection between locally uniformly joinable and coverable uniform spaces.

Findings

Identifies an error in Plaut's proof.

Clarifies the relationship between locally uniformly joinable and coverable spaces.

Abstract

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform covering maps which was motivated by the work of Berestovskii and Plaut. It is easy to see that a chain connected coverable uniform space is locally uniformly joinable. This paper points out an error in the attempt in Plaut's "An equivalent condition for a uniform space to be coverable" to prove that a locally uniformly joinable chain connected uniform space is coverable.