# directed set

A 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 residual if there is $a\in A$ such that $b\in B$ whenever $a\leq b$, and cofinal if for each $a\in A$ there is $b\in B$ such that $a\leq b$.

A directed set is sometimes called an upward-directed set. We may also define the dual notion: a downward-directed set (or 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.

 Title directed set Canonical name DirectedSet Date of creation 2013-03-22 12:54:00 Last modified on 2013-03-22 12:54:00 Owner yark (2760) Last modified by yark (2760) Numerical id 11 Author yark (2760) Entry type Definition Classification msc 06A06 Synonym upward-directed set Synonym upward directed set Related topic Cofinality Related topic AccumulationPointsAndConvergentSubnets Defines residual Defines cofinal Defines downward-directed set Defines downward directed set Defines filtered set