Minimal subcoverings of a cover with finite intersection

Question

Let $X\neq \emptyset$ be a set, and let ${\cal U}$ be a collection of subsets of $X$ such that

1. $\bigcup {\cal U} = X$, and
2. $U_1\neq U_2\in {\cal U}$ implies $|U_1\cap U_2| < \aleph_0$.

Is there ${\cal U}_0\subseteq {\cal U}$ such that

1. $\bigcup {\cal U}_0 = X$, and
2. if $U\in{\cal U}_0$ then $\bigcup \big({\cal U}_0 \setminus \{U\}\big) \ne X$ ?

Answer 1

No, consider the covering of naturals by initial segments.