Definition 1

The right (resp. left) uniformity on a topological group G is the uniformity for which a fundamental system of entourages is obtained by making correspond to each neighbourhood V of the identity element e, the set V$_d$ (resp. V$_s$) of pairs $(x, y)$ such that $yx^{-1} \in V$ (resp. $x^{-1}y \in V$).

If V runs through a fundamental system of neighbourhoods of e, the sets V$_d$ (resp. V$_s$) form a fundamental system of entourages of the right (resp. left) uniformity.

To each proposition on the topology of a uniform space there corresponds a proposition on the topology of a group; the translation is made according to Definition 1 and the formulae $V_d(x) = V.x,\ V_d(A) = V.A,\ V_s(x) = x.V,\ V_s(A) = A.V,$ which are immediate consequences of the definition. For example, if A is any non-empty subset of G then we have (Chapter II, § 1, no. 2, Corollary 1 to Proposition 2)

Again (Chapter II, § 1, no. 2, Corollary 3 to Proposition 2), every Hausdorff group is regular.

The right and left uniformities on a topological group are in general distinct (see Exercise 4). Obviously they coincide if the group is commutative, for then $V_d = V_s$; also they coincide if the group is compact (Chapter II, § 4, no. 1, Theorem 1).

In general, we shall denote by $G_s$ (resp. $G_d$) the uniform space obtained by giving the set G its left (resp. right) uniformity.