Maximal set with respect to the finite intersection property Let $X$ be a Hausdorff space. Let $\mathcal{D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. That means that $\mathcal{D}$ is a family of subsets of $X$ that has the finite intersection property such that any other family properly containing $\mathcal{D}$ does not have the finite intersection property. I would like to exhibit the following assertions:
(a) $x \in \overline{D} \ \forall D \in \mathcal{D} \iff$ every neighbourhood of $x$ belongs to $\mathcal{D}$. Which implication uses the maximality assumption?
(b) Let $D \in \mathcal{D}$. Show that: $A \supset D \Rightarrow A \in \mathcal{D}$
(c) If $X$ is $T_1$ there is no more than one point in $\displaystyle \bigcap_{D \in \mathcal{D}} \overline{D}$
I think that the reverse implication of $(a)$ is straightforward. I am only worrying about the forward one, which I assume is the one that uses maximality. I have had no luck with the others.
I realise that there is a conflict. In $(c)$ one must assume that $X$ is $T_1$ but that comes for free since we assume $X$ Hausdorff. Our professor told us that one should assume Hausdorff otherwise the statement is wrong, although he did not specify for which part. The book does not provide the Hausdorff condition.
 A: $\newcommand{\cl}{\operatorname{cl}}$HINTS: 
(a) Your suspicion that maximality of $\mathscr{D}$ is used for the forward implication is correct. Note that if $x\in\cl D$ for some set $D$, and $N$ is a nbhd of $x$, then $N\cap D\ne\varnothing$. Note also that if $\mathscr{D}$ is maximal, then for any finite subset $\mathscr{F}$ of $\mathscr{D}$ we must have $\bigcap\mathscr{F}\in\mathscr{D}$.
(b) You must again use maximality: show that if $A\supseteq D$ for some $D\in\mathscr{D}$, then $\mathscr{D}\cup\{A\}$ has the finite intersection property.
(c) This is false as stated. Let $\tau$ be the cofinite topology on $\Bbb N$, and let $\mathscr{T}$ be the family of non-empty open sets: $U\in\mathscr{T}$ if and only if $\Bbb N\setminus U$ is finite. Clearly $\mathscr{T}$ has the finite intersection property. Using Zorn’s lemma or one of its equivalents we can expand $\mathscr{T}$ to a family $\mathscr{D}\supseteq\mathscr{T}$ of subsets of $\Bbb N$ that is maximal with respect to having the finite intersection property. Every infinite subset of $\Bbb N$ is dense in the space, and every $D\in\mathscr{D}$ is infinite, so $\cl D=\Bbb N$ for every $D\in\mathscr{D}$, and therefore $$\bigcap_{D\in\mathscr{D}}\cl D=\Bbb N$$ contains more than one point. This is actually the part of the problem where you need the assumption that $X$ is Hausdorff: use that to show that if $x\ne y$, then $x$ has a nbhd $N$ such that $y\notin\cl N$.
A: For both (a) and (b), at least, you will get some mileage out of taking a set that is not in $\mathcal D$ and probing what happens when you add it to $\mathcal D$: since the larger collection does not have the finite intersection property (using maximality), you will learn that the additional set is disjoint from something having to do with $\mathcal D$. In the forward implication of part (a) (or rather its contrapositive), let this additional set be a neighborhood of $x$ not belonging to $\mathcal D$. In (b) (again its contrapositive), let the additional set be $A\notin\mathcal D$.
