"Uniform groups" (similar to topological groups)? Why have I heard about topological groups, but nothing about "uniform groups" (uniform spaces endowed with a group)?

Every topological group is automatically a uniform group. In particular $G$ becomes a uniform space if we define a subset $V$ of $G\times G$ to be an entourage if and only if it contains the set $\{ (x, y) : x⋅y^{−1} \in U \}$ for some neighborhood $U$ of the identity element of $G.$
The way to think about it is that uniform spaces have ways to compare neighborhoods at different points, and the group action on itself allows us to do that for $G$.
