"Uniform groups" (similar to topological groups)?

Question

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

Answer 1

A topological group is a uniform space in two ways... one by left translation, the other by right translation. Replace $x\cdot y^{-1}$ in Owen's answer by $y^{-1}\cdot x$ to switch. The two are equivalent in abelian groups, compact groups, and in many others. But not all.

I have heard it said that the notion of "uniform space" was originally given as a common generalization of "metric space" and "topological group".

Answer 2

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 $G$.