I have a list of questions concerning the properties of filters:
(1) If a finite subset of a poset is downward directed is it necessarily closed under finite intersection? At the very least, if said subset is downward directed, is the intersection of any two subsets of the subset contained in the subset?
(2) If a set is directed does it have a unique bound? For example, do all downward directed sets have a unique lower bound?
(3) Can a filter of neighborhoods of x, where x \in R, be conceived of as a family of intervals containing x and in which x is the only element in the intersection of all subsets in said family?
Thank you for your time.
(1) No. In the first place, in a general poset, there is no notion of intersection. Even if your poset is the set $P(X)$ of all subsets of a set $X$, so that "intersection" makes sense, downward-directedness does not guarantee anything about closure under intersections, even binary intersections.
(2) No. Anything below a lower bound of a set is another lower bound of that set. The greatest lower bound of a set is unique if it exists, but in general posets it need not exist. In $P(X)$, greatest lower bounds exist, but the greatest lower bound of a downward-directed set need not be a member of that set.
(3) A filter, being closed under supersets, won't consist only of intervals. In the real line, the neighborhood filter of a point $x$ consists of the open intervals containing $x$ and all the supersets of those intervals. $x$ is indeed the only point in the intersection of all those intervals.