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In [[mathematics]], a '''directed set''' (or a '''directed preorder''' or a '''filtered set''') is a nonempty [[Set (mathematics)|set]] ''A'' together with a [[reflexive relation|reflexive]] and [[transitive relation|transitive]] [[binary relation]] ≤ (that is, a [[preorder]]), with the additional property that every pair of elements has an [[upper bound]]:<ref>Kelley, p. 65.</ref> In other words, for any ''a'' and ''b'' in ''A'' there must exist a ''c'' in ''A'' with ''a'' ≤ ''c'' and ''b'' ≤ ''c''.
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Directed sets are a generalization of nonempty [[totally ordered set]]s, that is, all totally ordered sets are directed sets (contrast [[partially ordered sets|''partially'' ordered sets]] which need not be directed). 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]].
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== Equivalent definition ==
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In addition to the definition above, there is an equivalent definition. A '''directed set''' is a set ''A'' with a preorder such that every finite subset of ''A'' has an upper bound. In this definition, we take the upper bound of the [[empty set|empty subset]] to be any existing element of ''A'' and require that ''A'' be nonempty.


== Examples ==
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Examples of directed sets include:
* The set of [[natural number]]s '''N''' with the ordinary order ≤ is a directed set (and so is every [[total order|totally ordered set]]).
* Let '''D<sub>1</sub>''' and '''D<sub>2</sub>''' be directed sets. Then the Cartesian product set '''D<sub>1</sub> <math>\times</math> D<sub>2</sub>''' can be made into a directed set by defining (''n''<sub>1</sub>, ''n''<sub>2</sub>) ≤ (''m''<sub>1</sub>, ''m''<sub>2</sub>) if and only if ''n''<sub>1</sub> ≤ ''m''<sub>1</sub> and ''n''<sub>2</sub> ≤ ''m''<sub>2</sub>. In analogy to the [[product order]] this is the product direction on the Cartesian product.
* It follows from previous example that the set '''N''' <math>\times</math> '''N''' of pairs of natural numbers can be made into a directed set by defining (''n''<sub>0</sub>, ''n''<sub>1</sub>) ≤ (''m''<sub>0</sub>, ''m''<sub>1</sub>) if and only if ''n''<sub>0</sub> ≤ ''m''<sub>0</sub> and ''n''<sub>1</sub> ≤ ''m''<sub>1</sub>.
* If ''x''<sub>0</sub> is a [[real number]], we can turn the set '''R''' − {''x''<sub>0</sub>} into a directed set by writing ''a'' ≤ ''b'' if and only if <br>|''a'' − ''x''<sub>0</sub>| ≥ |''b'' − ''x''<sub>0</sub>|. We then say that the reals have been ''directed towards x<sub>0</sub>''. This is an example of a directed set that is not [[partial order|ordered]] (neither totally nor partially).
* A (trivial) example of a partially ordered set that is '''''not''''' directed is the set {''a'', ''b''}, in which the only order relations are ''a'' ≤ ''a'' and ''b'' ≤ ''b''. A less trivial example is like the previous example of the "reals directed towards x<sub>0</sub>" but in which the ordering rule only applies to pairs of elements on the same side of x<sub>0</sub>.
* If ''T'' is a [[topological space]] and ''x''<sub>0</sub> is a point in ''T'', we turn the set of all [[topological neighbourhood|neighbourhoods]] of ''x''<sub>0</sub> into a directed set by writing ''U'' ≤ ''V'' if and only if ''U'' contains ''V''.
** For every ''U'': ''U'' ≤ ''U''; since ''U'' contains itself.
** For every ''U'',''V'',''W'': if ''U'' ≤ ''V'' and ''V'' ≤ ''W'', then ''U'' ≤ ''W''; since if ''U'' contains ''V'' and ''V'' contains ''W'' then ''U'' contains ''W''.
** For every ''U'', ''V'': there exists the set ''U'' <math>\cap</math> ''V'' such that ''U'' ≤ ''U'' <math>\cap</math> ''V'' and ''V'' ≤ ''U'' <math>\cap</math> ''V''; since both ''U'' and ''V'' contain ''U'' <math>\cap</math> ''V''.
* In a [[poset]] ''P'', every lower closure of an element, i.e. every subset of the form {''a''| ''a'' in ''P'', ''a'' ≤''x''} where ''x'' is a fixed element from ''P'', is directed.


[[File:Directed set, but no join semi-lattice.gif|thumb|x100px|Witness]]
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== Contrast with semilattices ==


Directed sets are a more general concept than (join) semilattices: every [[semilattice|join semilattice]] is a directed set, as the join or least upper bound of two elements is the desired ''c''. The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the last bit 1 > 0), where {1000,0001} has three upper bounds but no ''least'' upper bound, cf. picture.
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== Directed subsets ==
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The order relation in a directed sets is not required to be [[antisymmetric relation|antisymmetric]], and therefore directed sets are not always [[partial order]]s. However, the term ''directed set'' is also used frequently in the context of posets. In this setting, a subset ''A'' of a partially ordered set (''P'',≤) 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 ''A'' is inherited from ''P''; 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 (mathematics)|filter]].
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Directed subsets are used in [[domain theory]], which studies [[directed complete partial order]]s.<ref>Gierz, p. 2.</ref> 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.
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* [[Filtered category]]
* [[Centered set]]
* [[Linked set]]
 
==Notes==
{{reflist}}
 
== References ==
 
* J. L. Kelley (1955), ''General Topology''.
 
* Gierz, Hofmann, Keimel, ''et al.'' (2003), ''Continuous Lattices and Domains'', Cambridge University Press. ISBN 0-521-80338-1.
 
[[Category:Mathematical relations]]
[[Category:Order theory]]
[[Category:General topology]]

Latest revision as of 17:54, 4 January 2015

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