The most important example of an $H$-group is the loop space $(\Omega Y,w_0)$ of any pointed space $(Y,y_0)$. Let $\mu:\Omega Y\times \Omega Y\to \Omega Y; \;\; \mu(\alpha,\beta)=\alpha \star\beta$,
where $\alpha \star\beta$ is the product of two loop $\alpha$ and $\beta$. To show $\mu$ is continuous, we use Theorem of exponential correspondence and Pasting lemma on $\Omega Y\times \Omega Y\times[0,\frac{1}{2}]$ and $\Omega Y\times \Omega Y\times[\frac{1}{2},1]$ as closed subsets of $\Omega Y\times \Omega Y\times I$. In fact we show that
$\Omega Y\times \Omega Y\times I\mathop \to \limits^{\mu\times 1}\Omega Y\times I\mathop \to \limits^{E}Y;\;\; (\alpha,\beta,t)\mapsto \alpha\star\beta(t),$ is continuous. $\mu$ is an H-multiplication. that is, the constant map $e:\Omega Y\to\Omega Y$ whose value is the constant loop $w_0$ is an $H$-unit (i.e. $\mu\circ (e,1)\simeq 1 \simeq \mu\circ (1,e)$.
We want a homotopy $H:\Omega Y\times I\to \Omega Y$ from $\mu\circ (1,e)$ to $1$. To do this, fixed $t\in I$ we define an $H_t:\Omega Y\times \{t\}\times I\to Y$ by $ H_t(w,t,s)= \begin{cases} w(\frac{2s}{t+1})& 0\leq s\leq \frac{t+1}{2}\\ y_0 & \frac{t+1}{2}\leq s\leq 1 \end{cases} .$
In a similar way, Pasting lemma on $\Omega Y\times\{t\}\times [0,\frac{t+1}{2}]$ and $\Omega Y\times\{t\}\times [\frac{t+1}{2},1]$ implies that $H_t$ is continuous for every $t\in I$. But now how we can show that $\bar H:\Omega Y\times I\times I\to Y$ by $ H(w,t,s)= \begin{cases} w(\frac{2s}{t+1})& 0\leq s\leq \frac{t+1}{2}\\ y_0 & \frac{t+1}{2}\leq s\leq 1 \end{cases} $ is continuous?
[1] Theorem of exponential correspondence: If $X$ is a locally compact hausdorff space and $Y$ and $Z$ are topological spaces, a map $g: Z\to Y^X$ is continuous if and only if $E\circ (g \times 1): Z\times X\to Y$ is continuous.
[1] E. H. Spanier, Algebraic topology, New York, McGraw-Hill, 1966.
