Exercise 7, § 3

a) Let G be a Hausdorff topological group. If there is a neighbourhood V_0 of e in G such that the symmetry $x \to x^{-1}$, considered as a mapping of $G_d$ into $G_d$, is uniformly continuous on $V_0$, show that G has a completion (show that the condition of Theorem 1 of § 4 is satisfied).

\* b) Show that if G is an infinite product of groups each identical with the topological group defined in Exercise 4, then G is complete but there is no neighbourhood of e on which $x \to x^{-1}$, considered as a mapping of $G_d$ into $G_d$, is uniformly continuous *.