For $X=\lbrace 0,\ldots,n-1\rbrace$, let $F\subseteq 2^X$ be a family of subsets of $X$ such that, for every $x\in X$, the singleton $\lbrace x\rbrace$ is the intersection of some elements of $F$. I am interested in the minimal families that have this property, in particular whether it is possible to have $|F|< n$. Can anyone (a) give an example where $|F|< n$, (b) provide an argument for why $|F|\ge n$, or (c) point me in the direction of some existing results.
Bonus question: For any $k< n$, the family $F= \lbrace \lbrace x,x+1,\ldots,x+k-1 \rbrace:x\in X \rbrace$, where addition is carried out modulo $n$, satisfies the stated condition (minimally) and contains exactly $n$ elements, so $|F|= n$ is always achievable. How does the structure of a general minimal family relate to these highly regular families? Is a minimal family always a disjoint union of some regular families?