Infinite family $\mathscr{A}\subseteq P(\omega)$ with criteria

Question

First part:

Prove that there's an infinite family $\mathscr{A}\subseteq P(\omega)$ such that:

1. $X \in \mathscr{A} \Rightarrow |X|=\aleph_0$
2. $(X,Y\in \mathscr{A} \wedge X\ne Y)\Rightarrow |X \cap Y|<\aleph_0$
3. $\forall Z\subseteq \omega$, if $|Z|=\aleph_0$ then $\exists X\in \mathscr{A}$ such that $|X\cap Z|=\aleph_0$.

Second part:

Prove that $\mathscr{A}$ is not countable.

I've started the first part with Zorn's lemma, but when I get a promised maximum of the set $S=\{\mathscr{A}\subseteq P(\omega) : \mathscr{A}\ meets\ the\ 3\ conditions\}$ with the $\subseteq$ relation I can't know weather $\mathscr{A}$ is finite or infinite.

Maybe the solution is through [ultra]filters, compactness theorem or transfinite induction instead of Zorn's lemma. What do you think?

Answer 1

No, Zorn's lemma is plenty.

The trick, however, is to start with an uncountable family $\scr B$ satisfying (1) and (2), and define the partial order $S$ of all $\scr A$ such that $\scr B\subseteq A$ and the conditions hold.

---

However, we can also notice the following trend: either $\scr A$ is finite, e.g. if it is $\scr A=\{\omega\}$, or $\scr A$ is uncountable.

Suppose that $\scr A$ was countably infinite, say $\{A_n\mid n<\omega\}$, let $a_{n,m}$ be the increasing enumeration of $A_n$ for $m<\omega$. Then define $x_n$ to be the least $a_{n,m}$ such that $a_{n,m}\notin A_k$ for all $k<n$. Using the finite intersections, we can prove that such $m$ exists, so $x_n$ is well-defined.

Now $X\cap A_n$ must be finite, so condition (3) must fail.

Answer 2

A topological argument: a family $\mathcal{A}$ that obeys 1. to 3. is called a maximal almost disjoint family in $\mathscr{P}(\omega)$. Almost disjoint means that two infinite sets have finite intersection. And the maximality follows from the last clause, as it implies that if $A$ is infinite and not in $\mathcal{A}$ it cannot be added to $\mathcal{A}$ without killing almost disjointness. Zorn applied the the poset of all such families (partially ordered by inclusion) yields, by standard arguments, the existence of an infinite such m.a.d.family (we merely extend any infinite pairwise disjoint family, say).

For any such a family we can construct a Mrówka $\Psi$-space denoted $\Psi(\mathcal{A})$ that has the following properties, proved from it being an infinite m.a.d. family (see here for construction and proofs):

• $\Psi(\mathcal{A})$ has the form $\Psi(\mathcal{A})= \omega \cup \{x_A: A \in \mathcal{A} \}$
• $\Psi(\mathcal{A})$ is locally compact, zero-dimensional, Hausdorff, separable, first-countable.
• $\Psi(\mathcal{A})$ is pseudocompact but not countably compact (and thus not normal, as a normal psudocompact space is countably compact). $\mathcal{A}$ not being countably compact is shown by showing that $\{x_A: A \in \mathcal{A}\}$ is closed and discrete in $\Psi(\mathcal{A})$, and this set is infinite.

If a countable m.a.d. family $\mathcal{A}$ existed, then $\Psi(\mathcal{A})$ would be a countable, first countable (so second countable) regular space and hence metrisable and normal, which it is not. Ergo, no such family can exist. Quite indirect, but Mrówka's spaces are one of my favourite examples.