Proposition 5

There exists a unique topology on $G^*$ compatible with its group structure for which the sets $V^*_{\varepsilon}$ form a basis of neighbourhoods of the identity element.

Let us verify that the sets $V_{\varepsilon}^*$ satisfy axioms (GV$'_I$), (GV$'_{II}$) and (GV$'_{III}$) of TG, III, p. 4.

Let V be a neighbourhood of $e$ in G and let $\varepsilon$ be a strictly positive real number. Let W be a neighbourhood of $e$ in G such that $W\cdot W\subset V$. Let $h$ and $h'$ be elements of $G^*$. If $t\in [0,1[$ is such that $h(t)h'(t)\notin V$, one has $h(t)\notin W$ or $h'(t)\notin W$. It follows that $p_V(hh')\leqslant p_W(h)+p_W(h')$. Consequently, $W^*_{\varepsilon /2}\cdot W^*_{\varepsilon /2}\subset V^*_{\varepsilon}$, which proves axiom (GV$'_I$).

Let W be a neighbourhood of $e$ in G such that $W^{-1}\subset V$. Then $(W^*_{\varepsilon})^{-1}=$ $(W^{-1})^*_{\varepsilon}\subset V^*_{\varepsilon}$, whence axiom (GV$'_{II}$).

Let $k$ be an element of $G^*$; since the function $k$ takes only a finite number of values, there exists a neighbourhood W of $e$ in G such that $k(t)Wk(t)^{-1}\subset V$ for all $t\in [0,1[$. Let $h$ then be an element of $G^*$. For $t\in [0,1[$, if $k(t)h(t)k(t)^{-1}\notin$ V, then $h(t)\notin W$. Consequently, $p_V(khk^{-1})\leqslant p_W(h)$. This proves that $kW^*_{\varepsilon}k^{-1}\subset V^*_{\varepsilon}$, whence axiom (GV$'_{III}$).