An equivalent condition for a uniform space to be coverable

TL;DR

This paper establishes a new equivalent condition for a uniform space to be coverable, linking it to the uniform openness of projections in the fundamental inverse system, and clarifies related properties.

Contribution

It introduces an equivalent condition for coverability in uniform spaces and corrects previous errors, connecting coverability with chain connectedness and uniform joinability.

Findings

Provides a concrete method to find covering entourage.

Corrects an error in previous literature.

Shows coverability is equivalent to chain connectedness and uniform joinability.

Abstract

We prove that an equivalent condition for a uniform space to be coverable is that the images of the natural projections in the fundamental inverse system are uniformly open in a certain sense. As corollaries we (1) obtain a concrete way to find covering entourage, (2) correct an error in [3] and (3) show that coverable is equivalent to chain connected and uniformly joinable in the sense of arXiv:0706.3937.