Uniformly joinable, locally uniformly joinable, and weakly chained uniform spaces

TL;DR

This paper explores various notions of uniform joinability in uniform spaces, providing examples and establishing equivalences between weak chainability and pointed 1-movability in metric continua.

Contribution

It introduces an example of a uniformly joinable space that is not locally uniformly joinable and links weak chainability to pointed 1-movability.

Findings

Example of a uniformly joinable but not locally uniformly joinable space

Weakly chained metrizable uniform spaces are locally uniformly joinable

Weak chainability is equivalent to pointed 1-movability in metric continua

Abstract

Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local uniform joinability for uniform spaces when developing their theory of generalized uniform covering maps which was motivated by a paper by Berestovskii and Plaut. (Local) uniform joinability can be thought of as analogous to (local) path connectedness. A chain connected locally uniformly joinable uniform space is uniformly joinable. This note gives an example of a metric space that is uniformly joinable but not locally uniformly joinable. Plaut recently defined the concept of a weakly chained uniform space. We show that a weakly chained metrizable uniform space is locally uniformly joinable. Since local uniform joinability is equivalent to pointed 1-movability for metric continua, we find that weak chainability is equivalent to pointed 1-movability for such spaces.