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.