Given a uniform space in the entourage sense, define a cover P to be uniform if there is some entourage "U" such that for each "x" ∈ "X", there is an "A" ∈ P such that "U"["x"] ⊆ "A". These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of ⋃{"A" × "A" : "A" ∈ P}, as P ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other.