The author says,
"If $X$ is a set and $\mathbb A$ be the superset consisting of all collections $\mathscr B$ of subsets of $X$ such that $\mathscr A \subset \mathscr B$ and $\mathscr B$ has the countable intersection property. and $\mathbb A$ has strict partial order by proper inclusion.
If $\mathscr A$ is a collection of subsets of $X$ having the countable intersection property, then there is a collection $\mathscr D$ of subsets of $X$ such that $\mathscr D\subset \mathscr A$ and $\mathscr D$ is maximal with respect to the countable intersection property." is wrong.
Actually this is to try to generalize some lemma which was used to prove Tychonoff theorem. The lemma is "If $X$ is a set and $\mathscr A$ is a collection of subsets of $X$ having the finite intersection property, then there is a collection $\mathscr D$ of subsets of $X$ such that $\mathscr D\subset \mathscr A$ and $\mathscr D$ is maximal with respect to the finite intersection property."
I guess generalizing this lemma to the countable case fails because the upper bound on $\mathbb B$ which is given subsuperset of $\mathbb A$ that is simply ordered by proper inclusion is not an element of $ \mathbb A$. So we cannot apply Zorn's lemma.
But at this point, I couldn't show my guess is right.
Is my guess right? if not, please give me a little hint so that I could proceed by myself.