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

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.