Exercise 4, § 3

On every topological group $G$ there exists a family of left-invariant pseudometrics such that the uniformity defined by this family is the same as the left uniformity of $G$.

2) Let $G$ be a topological group whose left and right uniformities coincide. Show that this unique uniformity can be defined by a family of pseudometrics which are simultaneously left-and right-invariant (using Exercise 3 of Chapter III, § 3, show that if $V$ is any neighbourhood of the identity element $e$ of $G$, then $V_0 = \bigcap_{x \in G} xVx^{-1}$ is a neighbourhood of $e$).

3) Let $G$ be a topological group, let $\{f_i\}$ be a saturated family of left-invariant pseudometrics on $G$ which define the left uniformity of $G$, and let $g_i(x) = f_i(e, x)$. Let $H$ be a normal closed subgroup of $G$, and for each coset $x \in G/H$ let $h_i(x) = \inf_{x \in x} g_i(x)$; show that, if $f_i(x, y) = h_i(x^{-1}y)$, the $f_i$ form a family of left-invariant pseudometrics on $G/H$ which define the left uniformity of $G/H$ (argue as in Remark 2 of no. 1').

4) Show that the topology of a valued division ring is locally retrobounded (Chapter III, § 6, Exercise 22).