Let $X$ be a $T_1$ space. Let $\mathfrak {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that there is at most one point belonging to $\bigcap_{D \in \mathfrak{D}} \bar D$
Here's my attempt. Suppose there're two points $x$, $y$ both belonging to $\bigcap_{D \in \mathfrak{D}} \bar D$. By $T_1$ axiom, there is one open set $A$, such that $x \in A$, while $y \notin A$. The maximality of $\mathfrak D$ implies every neighborhood of $x$ and $y$ belongs to $\mathfrak D$, and so is $A$. Thus every neighborhood of $y$ intersects with $A$, which shows that $y\in \bar A$.
I don't know how to proceed.
EDIT:This is an excercise from James Munkres' textbook Topology(2ed), page 235. However, I can't find a problem in K.Gosh's answer. Maybe something is wrong in this problem, or my paraphrasing of it.