Cardinality of the set of minimal sets of a collection with cardinality $\aleph_0$

Question

Let $X$ be a set and $F$ be a collection of subsets of $X$ such that $\vert F \vert = \aleph_0$ and let $F^*$ be the smallest collection of subsets of $X$ closed under intersection and complement (w.r.t. $X$) containing $F$. Given a set $s \in F^*$, call it '$F$-minimal' whenever $\neg \exists s' \in F$ such that $s' \subset s$. Let $M$ be the collection of all $F$-minimal sets. Do we have $\vert M \vert \leq \aleph_0$?

Answer 1

Let $X$ be any countably infinite set, and let $\mathscr{F}=\big\{X\setminus\{x\}:x\in X\big\}$. Then $\mathscr{F}^*=\wp(X)\setminus\{X\}$, and every element of $\mathscr{F}^*\setminus\mathscr{F}$ is $\mathscr{F}$-minimal. Clearly $\mathscr{F}^*\setminus\mathscr{F}$ is uncountable.