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 $\bigcup \big({\cal U}\_0 \setminus \{U\}\big) \ne X$
?



Minimal subcoverings of a cover with finite intersection
No, consider the covering of naturals by initial segments.


