Editing Directed set (section)

{{Short description|Mathematical ordering with upper bounds}}
In [[mathematics]], a '''directed set''' (or a '''directed preorder''' or a '''filtered set''') is a nonempty [[Set (mathematics)|set]] <math>A</math> together with a [[Reflexive relation|reflexive]] and [[Transitive relation|transitive]] [[binary relation]] <math>\,\leq\,</math> (that is, a [[preorder]]), with the additional property that every pair of elements has an [[upper bound]].{{sfn|Kelley|1975|pp=65}} In other words, for any <math>a</math> and <math>b</math> in <math>A</math> there must exist <math>c</math> in <math>A</math> with <math>a \leq c</math> and <math>b \leq c.</math> A directed set's preorder is called a '''direction'''.

The notion defined above is sometimes called an '''{{visible anchor|upward directed set}}'''. A '''{{visible anchor|downward directed set}}''' is defined analogously,<ref>{{cite book|author=Robert S. Borden|title=A Course in Advanced Calculus|year=1988|publisher=Courier Corporation|isbn=978-0-486-15038-3|page=20}}</ref> meaning that every pair of elements is bounded below.<ref name="Brown-Pearcy">{{cite book|author1=Arlen Brown|author2=Carl Pearcy|title=An Introduction to Analysis|url=https://archive.org/details/introductiontoan0000brow|url-access=registration|year=1995|publisher=Springer|isbn=978-1-4612-0787-0|page=[https://archive.org/details/introductiontoan0000brow/page/13 13]}}</ref>{{efn|That is, for any <math>a</math> and <math>b</math> in <math>A</math> there must exist <math>c</math> in <math>A</math> with <math>c \leq a</math> and <math>c \leq b</math>.}} 
Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward.<ref name="CarlHeikkilä2010">{{cite book|author1=Siegfried Carl|author2=Seppo Heikkilä|title=Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory|year=2010|publisher=Springer|isbn=978-1-4419-7585-0|pages=77}}</ref>

Directed sets are a generalization of nonempty [[totally ordered set]]s. That is, all totally ordered sets are directed sets (contrast [[Partially ordered sets|{{em|partially}} ordered sets]], which need not be directed). [[Join-semilattice]]s (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, [[Lattice (order)|lattice]]s are directed sets both upward and downward.

In [[topology]], directed sets are used to define [[Net (topology)|nets]], which generalize [[sequence]]s and unite the various notions of [[Limit (mathematics)|limit]] used in [[Mathematical analysis|analysis]]. Directed sets also give rise to [[direct limit]]s in [[abstract algebra]] and (more generally) [[category theory]].