Editing Directed set (section)

==Directed subsets==
The order relation in a directed set is not required to be [[Antisymmetric relation|antisymmetric]], and therefore directed sets are not always [[partial order]]s. However, the term {{em|directed set}} is also used frequently in the context of posets. In this setting, a subset <math>A</math> of a partially ordered set <math>(P, \leq)</math> is called a '''directed subset''' if it is a directed set according to the same partial order: in other words, it is not the [[empty set]], and every pair of elements has an upper bound. Here the order relation on the elements of <math>A</math> is inherited from <math>P</math>; for this reason, reflexivity and transitivity need not be required explicitly.

A directed subset of a poset is not required to be [[Lower set|downward closed]]; a subset of a poset is directed if and only if its downward closure is an [[Ideal (order theory)|ideal]]. While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a [[Filter (set theory)|filter]].

Directed subsets are used in [[domain theory]], which studies [[Complete partial order|directed-complete partial order]]s.{{sfn|Gierz|Hofmann|Keimel|Lawson|2003|p=2}} These are posets in which every upward-directed set is required to have a [[least upper bound]]. In this context, directed subsets again provide a generalization of convergent sequences.{{explain|reason=Again? Convergent sequences are never mentioned in this article.|date=December 2020}}