In the beginning of chapter 4 of Dr. Pete Clark's convergence notes:
http://math.uga.edu/~pete/convergence.pdf
Theorem 4.1 (page 13) asserts the equivalence of 5 conditions. After making a nice observation involving De Morgan's law and applying a previous proposition, the problem of proof is reduced to showing the equivalence of the following two statements (extracted from Theorem 4.1 referenced above):
Let $X$ be a topological space.
(b) Every net in $X$ has a limit point.
(e) For every family $\{F_{\alpha}\}_{\alpha\in J}$ of closed subsets with the finite intersection property (the finite intersection of any finite subcollection is non-empty), $\cap_{\alpha\in J}F_{\alpha}\neq \phi$.
For the $(b)\Rightarrow (e)$ direction, it is stated that such a family $F:=\{F_{\alpha}\}_{\alpha\in J}$ is directed under reverse set inclusion (this is used to create a net to which (b) is applied).
I cannot see this.
Of course it is a partially ordered set. But I do not think it needs to be directed. For example, if $X = [0,1]$, $F_{0} = [0,\frac{1}{2}]$, and $F_{1} = [\frac{1}{2},1]$ then $F:=\{F_{0},F_{1}\}$ satisfies the finite intersection property but is not directed. That is, there is no $A\in F$ such that $A\subset F_{0}$ and $A\subset F_{1}$.
Is there a way to argue away this type of case so that we can safely assume that we have a directed set?