Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a *cover* if $\bigcup {\cal C} = X$. A subset $D\subseteq X$ is a *choice set* for ${\cal C}$ if $|D\cap c| = 1$ for all $c\in C$. As Bjørn Kjos-Hanssen pointed out, choice sets do not always exist.

What is an example of a set $X$ and a cover ${\cal S}$ such that for no cover ${\cal S}_0 \subseteq {\cal S}$ there is a choice set for ${\cal S}_0$?