Set Theory and Zorn Lemma

Question

Prove that there is a set $B\subseteq P(\mathbb N)$ such that for all $n\in \mathbb N:\mathbb N- n\in B$, every finite intersection of elements in B is not empty and for all $C\subseteq\mathbb N$ such that $C\not\in B$ there are $n$ elements in $B$ such that: $B_1\cap B_2\cap\dots B_n\cap C=\emptyset$. I would like a hint for this problem. I think I need to use Zorn's lemma on a partial order set with the $\subset$ partial order but I can't find the set.

Answer 1

To use Zorn's lemma one wants to later exploit maximality. One can easily construct a set which satisfies all but the last one. For example all the co-finite subsets of $\Bbb N$.

So the maximality should be used to prove the last property.

In that case, the other properties should define the partial order $(A,\leq)$. We start by the naive approach, using $\subseteq$ as the ordering, and see if we can bound chains using their union. In case that fails, we will see what can we do about it.

Take $A$ to be all the subsets $B\subseteq\mathcal P(\Bbb N)$ with the first two properties being true. And order them by inclusion. Suppose that $\cal C$ is a chain in $A$, and consider $\bigcup\cal C$. It is not hard to see that it indeed have the wanted properties.

By Zorn's lemma there is a maximal element $B$. Now the usual maximality arguments can be used to show that the last property holds for $B$.

Answer 2

Let $\mathscr{C}$ be the class consisting of all sets $\mathcal{B}\subseteq P(\mathbb{N})$ satisfying:

• $\forall n\in\mathbb{N}$, $\mathbb{N}\setminus\left\{n\right\}\in \mathcal{B}$;
• every finite intersection of elements of $\mathcal{B}$ is non-empty.

Apply Zorn's lemma to $\mathscr{C}$ (ordered by inclusion). Any maximal element of $\mathscr{C}$ should do the job.