Proposition 6

The space $G^*$ is contractible and locally contractible at each of its points.

For $h\in G^*$ and $t\in \mathbf{I}$, let us denote by $\sigma (h, t)$ the mapping of $[0,1[$ into G defined by $\sigma (h, t)(x) =h(x)$ if $0\leqslant x < t$ and $\sigma (h, t) =e$ otherwise.

Let us show that the mapping $\sigma : G^*\times \mathbf{I}\rightarrow G^*$ resulting is continuous. Let in fact $k\in G^*,u\in \mathbf{I}$, let V be a neighbourhood of $e$ in G and let $\varepsilon$ be a strictly positive real number. The element $\sigma (h, t)\sigma (k, u)^{-1}$ of $G^*$ is the mapping $f$ of $[0,1[$ into G defined by

$h(x)k(x)^{-1}$ if $0\leqslant x <$ min($t, u$) ;

$h(x)$ if $u\leqslant x < t$;

$k(x)^{-1}$ if $t\leqslant x < u$;

$e$ otherwise.

Consequently,

in other words, for $\sigma (h, t)\in V^*_{\varepsilon}\sigma (k, u)$, it is enough that one have $|t-u|\leqslant \frac{\varepsilon}{2}$ and $h\in V^*_{\varepsilon /2}k$, which proves the continuity of $\sigma$ at $(k, u)$.

For every $h\in G^*,\sigma (h,0)$ is the constant mapping with image $\{e\}$, whereas $\sigma (h,1) =h$. Moreover, $\sigma (e, t) =e$ for every $t\in \mathbf{I}$. Consequently, $\sigma$ is a pointed homotopy at $e\in G^*$ joining the constant mapping with image $\{e\}$ to the identity mapping of $G^*$. This proves that $G^*$ is contractible at $e^*$.

Moreover, for every neighbourhood V of $e$ in G and every real number $\varepsilon >0$, one has $\sigma (V^*_{\varepsilon}\times \mathbf{I})\subset V^*_{\varepsilon}$. Hence $V^*_{\varepsilon}$ is also contractible at $e^*\in$ $G^*$, so that $G^*$ is locally contractible at $e^*$.

Since $G^*$ is a topological group, it is contractible and locally contractible at each of its points.

Let $\iota$ be the mapping of G into $G^*$ which associates to $g\in G$ the constant mapping with image $\{g\}$ of $[0,1[$ into G. The mapping $\iota$ is an injective homomorphism of groups. Let V be a neighbourhood of $e$ in G and let $\varepsilon$ be a strictly positive real number. One has $\overset{-1}{\iota}(V^*_{\varepsilon}) = V$ if $\varepsilon \leqslant 1$ and $\overset{-1}{\iota}(V^*_{\varepsilon}) = G$ otherwise. The inverse image of a neighbourhood of the identity element of $G^*$ is a neighbourhood of the identity element of G, whence the continuity of $\iota$. Moreover, $\iota (V) = V^*_1\cap \iota (G)$ for every neighbourhood V of $e$ in G. Hence $\iota$ defines an isomorphism of topological groups of G onto its image.