Example 2

Let X be a topological space and $x$ a point of X. Endowed with the juxtaposition of loops at $x$, the pointed space $(\Omega_x(X), e_x)$ is a Hopf space. In fact, first one has $e_x*e_x=e_x$. On the other hand, let $\psi :\mathbf{I}\rightarrow \mathbf{I}$ be the function defined by $\psi (t) = 2t$ for $0\leqslant t\leqslant \frac{1}{2}$ and $\psi (t) = 1$ for $\frac{1}{2}\leqslant t\leqslant 1$ (cf. III, p. 291), and let $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow \mathbf{I}$ be a strict homotopy connecting $\psi$ with Id$_{\mathbf{I}}($III, p. 289, example). Then, for every loop $c\in \Omega_x(X)$, the mapping $c\circ \sigma :\mathbf{I}\times \mathbf{I}\rightarrow X$ is a strict homotopy connecting $c*e_x$ with $c$ in $\Omega_x(X)$.

Let $\tau$ be the mapping $\Omega_x(X)\times \mathbf{I}\rightarrow \Omega_x(X)$ defined by $\tau (c, s)(t) =$ $c\circ \sigma (s, t)$. Let us now prove that $\tau$ is continuous. By proposition 1 of III, p. 257, it is enough to show that the mapping $(c, s, t)\mapsto c(\sigma (s, t))$ of $\Omega_x(X)\times \mathbf{I}\times \mathbf{I}$ into X is continuous, or again, since $\mathbf{I}\times \mathbf{I}$ is compact, that the mapping $c\mapsto c\circ \sigma$ of $\mathscr{C}_c(\mathbf{I}; X)$ into $\mathscr{C}_c(\mathbf{I}\times \mathbf{I}; X)$ is continuous. This last assertion then follows from the lemma, I, p. 132, b). Hence the mapping $\tau$ is a homotopy connecting the mapping $c\mapsto c*e_x$ with the identity mapping of $\Omega_x(X)$. One has $\tau (e_x, s)(t) =x$ for all $s,t\in \mathbf{I}$; therefore $\tau$ is a homotopy pointed at $e_x$. One reasons analogously for the mapping $c\mapsto e_x*c$.