Exercise 30, § 5

Let $G$ be a group and let $d$ be a metric on $G$ such that, with respect to the topology $\mathscr{T}$ defined by $d$ on $G$, the mappings $y \to x_0y$ and $y \to yx_0$ of $G$ into $G$ are continuous for each $x_0 \in G$.

a) Show that if $G$ is complete with respect to the topology $\mathscr{T}$, then the mapping $(x, y) \to xy$ is continuous on $G \times G$ (use Exercise 23).

b) Suppose in addition that $\mathscr{T}$ has a countable base. Show that the mapping $x \to x^{-1}$ is then continuous on $G$, i.e., that the topology $\mathscr{T}$ is compatible with the group structure of $G$. [In the group $G \times G$, endowed with the topology $\mathscr{T} \times \mathscr{T}$, consider the set $F$ of all points $(x, x^{-1})$ where $x \in G$; $F$ is closed in $G \times G$, and the law of composition $(x, x^{-1})(y, y^{-1}) = (xy, y^{-1}x^{-1})$ defines a group structure on $F$; show that the topology induced on $F$ by that of $G \times G$ is compatible with this group structure by using $a$]. Then prove, by arguing as in Exercise 28 b), that the projection $pr_1$ of $F$ onto $G$ is bicontinuous.]