A uniform cover is an open cover in which enough of the subsets have a uniform minimal size. This doesn't make sense in (say) an arbitrary topological space, but it does make sense in a uniform space (or a uniform locale). In fact, it's possible to define a uniform space by specifying which covers are uniform.
Let be a (pseudo)metric space. A collection of subsets of is a uniform cover if:
If is a topological group, then we have both left-uniform and right-uniform covers, depending on whether we use or in the definition above. (I'm not sure what the convention is, if any, on which is left and which is right.)
Sometimes the notion of uniform cover is taken as axiomatic in the definition of uniform space. But if we define a uniform space in terms of entourages, then we have:
Let be a uniform space. A collection of subsets of is a uniform cover if:
This definition subsumes both (pseudo)metric spaces and topological groups (although a nonabelian topological group has two uniform structures, corresponding to the two notions of uniform cover).