Can you please help me solve this problem? I am completely lost. Thanks in advance.
Suppose that X is a space that $X=U \cup V$ with U, V open subsets.
Show that if $f$ is a path in X then [f] can be expressed as
$[f]=[f_1][f_2]...[f_q]$
with each $f_j$ being either a path in $U$ or a path in $V$.
This works for any open cover $(U_i)_i$. Let $V_i$ denote the preimage of $U_i$ under the path $f$. As $[0,1]$ is a compact metric space, you can apply Lebesgue's number lemma to find a Lebesgue number, that is a positive $\delta$ such that each interval $[x,x+\delta]$ is contained in a $V_i$. If you divide $[0,1]$ into $q$ equidistant intervals such that $q>1/\delta$, then the restrictions of $f$ to these intervals give the desired paths.