An equivalent condition for a uniform space to be coverable

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 [V. Berestovskii, C. Plaut, Uniform universal covers of uniform spaces, Topology Appl. 154 (2007) 1748–1777], and (3) show that coverability is equivalent to chain connectedness and uniform joinability in the sense of [N. Brodskiy, J. Dydak, B. Labuz, A. Mitra, Rips complexes and universal covers in the uniform category, preprint arXiv:0706.3937.