TITLE: Maximal set with respect to the finite intersection property
QUESTION [7 upvotes]: 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.

REPLY [7 votes]: $\newcommand{\cl}{\operatorname{cl}}$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 $y\notin\cl N$.