Assume that $I$ is a compact interval in $\mathbb R$. Does the following statement (I hope true) follows from compactness or from connectedness of $I$?
For arbitrary family of open in $\mathbb R$ subsets $U_s$, $s \in S$, if $I\subset \bigcup_{s \in S} U_i$ then there are $n \in \mathbb N$ and $s_1, \ldots, s_n\in S$ such that $I\subset U_{s_1} \cup... \cup U_{s_n}$, and $U_{s_i} \cap U_{s_{i+1}}\neq \emptyset$ for $i=1,...,n-1$.