directed set DirectedSet 2002-08-01 11:28:41 2007-02-04 11:25:17 Definition residual cofinal downward-directed set downward directed set filtered set \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{property} A \emph{directed set} is a partially ordered set $(A, \leq)$ such that whenever $a,b\in A$ there is an $x\in A$ such that $a\leq x$ and $b\leq x$. A subset $B\subseteq A$ is said to be \emph{residual} if there is $a\in A$ such that $b\in B$ whenever $a\leq b$, and \emph{cofinal} if for each $a\in A$ there is $b\in B$ such that $a\leq b$. A directed set is sometimes called an \emph{upward-directed set}. We may also define the dual notion: a \emph{downward-directed set} (or \emph{filtered set}) is a partially ordered set $(A, \leq)$ such that whenever $a,b\in A$ there is an $x\in A$ such that $x\leq a$ and $x\leq b$. Note: Many authors do not require $\leq$ to be antisymmetric, so that it is only a pre-order (rather than a partial order) with the given property. Also, it is common to require $A$ to be non-empty.