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.