Countable refinement of intersections in $\mathbb{N}$

Question

So here’s a fairly elementary problem I came across but which I have trouble with solving: suppose $\mathcal{C} \subset P(\mathbb{N})$. Prove there is a countable $\mathcal{C}_0 \subset \mathcal{C}$ such that $$\cap_{A \in \mathcal{C}}A=\cap_{A \in \mathcal{C_0}}A.$$ I think one needs to reason by contradiction and if this is not the case $\mathbb{N}$ would not be countable, but I can’t really make it precise. Any ideas?

Answer 1

Let $D:=\{\mathbb N-A\mid A\in C\}$. Notice that your problem is equivalent to proving that there is a countable $D_0\subseteq D$ such that $$\bigcup_{B\in D_0}B=\bigcup_{B\in D}B.$$ Can you construct such $D_0$?