Proposition 7

Suppose that G is a metrizable topological group. Then the topological group $G^*$ is metrizable. More precisely, if $d$ is a bounded distance on G which defines its topology, the topology of $G^*$ is defined by the distance $d^*$.

Let $d$ be a distance on G which defines its topology. Then, the mapping $d'$ given by $d'(h, h') =$ inf($d(h, h'),1$) is a bounded distance on G which also defines its topology (TG, IX, p. 3). It is therefore enough to prove that $d^*$ defines the topology of $G^*$ under the assumption that $d$ is bounded.

Let V be a neighbourhood of $e$ in G and let $\varepsilon$ be a strictly positive real number. Let $\delta$ be a strictly positive real number such that V contains the ball $B(e, \delta )$. Let $h\in G^*$ and let $(t_i)_{0\leqslant i\leqslant n}$ be a sequence adapted to $h$. Then,

$d(h(t_{i-1}),e)\geqslant \delta$

Consequently, the ball $B(e^*, \varepsilon \delta )$ in $G^*$ is contained in $V_{\varepsilon}^*$.

Conversely, let $\delta$ be a strictly positive real number and let Δ be a strictly positive upper bound for the diameter of G. Let V be a neighbourhood of $e$ in G contained in the ball $B(e, \delta /2)$. For a function $h\in G^*$ and a sequence $(t_i)_{0\leqslant i\leqslant n}$ adapted to $h$, one has

= $\sum_{i=1}^n(t_i-t_{i-1})d(h(t_{i-1}), e)$

$d(h(t_{i-1}),e)\leqslant \delta /2$

+ $\sum_{i=1}^n(t_i-t_{i-1})d(h(t_{i-1}), e)$

$d(h(t_{i-1}),e)>\delta /2$ $\leqslant \frac{\delta}{2}+ \Delta \sum_{i=1}^n(t_i-t_{i-1})$ $h(t_{i-1})\notin V$

The preceding inequality implies that, for every element $h$ of $V_{\delta /2\Delta}^*$, one has $d^*(h, e^*)\leqslant \delta$. Consequently, every ball of $G^*$ for the distance $d^*$ contains a neighbourhood of the identity element.