TITLE: Every loop space $(\Omega Y,w_0)$ has the structure of an $H$-group.
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: This is just the pasting lemma: you are pasting two continuous maps, one on $\Omega Y\times \{(s,t)\in I\times I:0\leq s\leq\frac{t+1}{2}\}$ and one on $\Omega Y\times\{(s,t)\in I\times I:\frac{t+1}{2}\leq s\leq 1\}$.  These are closed subsets of $\Omega Y\times I\times I$ whose union is the whole space, and the continuous maps agree on the intersection, so they paste to give a continuous map on all of $\Omega Y\times I\times I$.