In mathematics, a directed set is a set equipped with a transitive, reflexive relation under which each pair of elements is bounded above. In other words, a set A equipped with such a relation--call it  "≤"--is directed if whenever x and y are in A there exists an element z of A which satisfies both x ≤ z and y ≤ z.

For example, the set of finite sets of integers, equipped with the partial order given by set inclusion, is directed: if X and Y are finite sets of integers, then their union Z is another finite set which contains each of them.