Editing Directed set (section)

==Examples==

The set of [[natural number]]s <math>\N</math> with the ordinary order <math>\,\leq\,</math> is one of the most important examples of a directed set. Every [[Total order|totally ordered set]] is a directed set, including <math>(\N, \leq),</math> <math>(\N, \geq),</math> <math>(\Reals, \leq),</math> and <math>(\Reals, \geq).</math>

A (trivial) example of a partially ordered set that is '''{{em|not}}''' directed is the set <math>\{a, b\},</math> in which the only order relations are <math>a \leq a</math> and <math>b \leq b.</math> A less trivial example is like the following example of the "reals directed towards <math>x_0</math>" but in which the ordering rule only applies to pairs of elements on the same side of <math>x_0</math> (that is, if one takes an element <math>a</math> to the left of <math>x_0,</math> and <math>b</math> to its right, then <math>a</math> and <math>b</math> are not comparable, and the subset <math>\{ a, b \}</math> has no upper bound).

===Product of directed sets===

Let <math>\mathbb{D}_1</math> and <math>\mathbb{D}_2</math> be directed sets. Then the [[Cartesian product]] set <math>\mathbb{D}_1 \times \mathbb{D}_2</math> can be made into a directed set by defining <math>\left(n_1, n_2\right) \leq \left(m_1, m_2\right)</math> if and only if <math>n_1 \leq m_1</math> and <math>n_2 \leq m_2.</math> In analogy to the [[product order]] this is the product direction on the Cartesian product.  For example, the set <math>\N \times \N</math> of pairs of natural numbers can be made into a directed set by defining <math>\left(n_0, n_1\right) \leq \left(m_0, m_1\right)</math> if and only if <math>n_0 \leq m_0</math> and <math>n_1 \leq m_1.</math>

===Directed towards a point===

If <math>x_0</math> is a [[real number]] then the set <math>I := \R \backslash \lbrace x_0 \rbrace</math> can be turned into a directed set by defining <math>a \leq_I b</math> if <math>\left|a - x_0\right| \geq \left|b - x_0\right|</math> (so "greater" elements are closer to <math>x_0</math>). We then say that the reals have been '''directed towards <math>x_0.</math>''' This is an example of a directed set that is {{em|neither}} [[Partial order|partially ordered]] nor [[Total order|totally ordered]]. This is because [[Antisymmetric relation|antisymmetry]] breaks down for every pair <math>a</math> and <math>b</math> equidistant from <math>x_0,</math> where <math>a</math> and <math>b</math> are on opposite sides of <math>x_0.</math> Explicitly, this happens when <math>\{a, b\} = \left\{x_0 - r, x_0 + r\right\}</math> for some real <math>r \neq 0,</math> in which case <math>a \leq_I b</math> and <math>b \leq_I a</math> even though <math>a \neq b.</math> Had this preorder been defined on <math>\R</math> instead of <math>\R \backslash \lbrace x_0 \rbrace</math> then it would still form a directed set but it would now have a (unique) [[greatest element]], specifically <math>x_0</math>; however, it still wouldn't be partially ordered. This example can be generalized to a [[metric space]] <math>(X, d)</math> by defining on <math>X</math> or <math>X \setminus \left\{x_0\right\}</math> the preorder <math>a \leq b</math> if and only if <math>d\left(a, x_0\right) \geq d\left(b, x_0\right).</math>

===Maximal and greatest elements===

An element <math>m</math> of a preordered set <math>(I, \leq)</math> is a ''[[Maximal and minimal elements|maximal element]]'' if for every <math>j \in I,</math> <math>m \leq j</math> implies <math>j \leq m.</math>{{efn|This implies <math>j = m</math> if <math>(I, \leq)</math> is a [[partially ordered set]].}}
It is a ''[[Greatest element and least element|greatest element]]'' if for every <math>j \in I,</math> <math>j \leq m.</math>

Any preordered set with a greatest element is a directed set with the same preorder. 
For instance, in a [[poset]] <math>P,</math> every [[Upper set#Upper closure and lower closure|lower closure]] of an element; that is, every subset of the form <math>\{a \in P : a \leq x\}</math> where <math>x</math> is a fixed element from <math>P,</math> is directed.

Every maximal element of a directed preordered set is a greatest element.  Indeed, a directed preordered set is characterized by equality of the (possibly empty) sets of maximal and of greatest elements.

===Subset inclusion===

The [[subset inclusion]] relation <math>\,\subseteq,\,</math> along with its [[Duality (order theory)|dual]] <math>\,\supseteq,\,</math> define [[partial order]]s on any given [[family of sets]]. 
A non-empty [[family of sets]] is a directed set with respect to the partial order <math>\,\supseteq\,</math> (respectively, <math>\,\subseteq\,</math>) if and only if the intersection (respectively, union) of any two of its members contains as a subset (respectively, is contained as a subset of) some third member. 
In symbols, a family <math>I</math> of sets is directed with respect to <math>\,\supseteq\,</math> (respectively, <math>\,\subseteq\,</math>) if and only if 
:for all <math>A, B \in I,</math> there exists some <math>C \in I</math> such that <math>A \supseteq C</math> and <math>B \supseteq C</math> (respectively, <math>A \subseteq C</math> and <math>B \subseteq C</math>) 
or equivalently, 
:for all <math>A, B \in I,</math> there exists some <math>C \in I</math> such that <math>A \cap B \supseteq C</math> (respectively, <math>A \cup B \subseteq C</math>).

Many important examples of directed sets can be defined using these partial orders. 
For example, by definition, a [[Filter (set theory)|{{em|prefilter}}]] or {{em|filter base}} is a non-empty [[family of sets]] that is a directed set with respect to the [[partial order]] <math>\,\supseteq\,</math> and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would then be a [[Greatest element and least element|greatest element]] with respect to <math>\,\supseteq\,</math>). 
Every [[Pi-system|{{pi}}-system]], which is a non-empty [[family of sets]] that is closed under the intersection of any two of its members, is a directed set with respect to <math>\,\supseteq\,.</math> Every [[Dynkin system|λ-system]] is a directed set with respect to <math>\,\subseteq\,.</math> Every [[Filter (set theory)|filter]], [[Topology (structure)|topology]], and [[σ-algebra]] is a directed set with respect to both <math>\,\supseteq\,</math> and <math>\,\subseteq\,.</math>

====Tails of nets====

By definition, a {{em|[[Net (mathematics)|net]]}} is a function from a directed set and a [[Sequence (mathematics)|sequence]] is a function from the natural numbers <math>\N.</math> Every sequence canonically becomes a net by endowing <math>\N</math> with <math>\,\leq.\,</math>

If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is any [[Net (mathematics)|net]] from a directed set <math>(I, \leq)</math> then for any index <math>i \in I,</math> the set <math>x_{\geq i} := \left\{x_j : j \geq i \text{ with } j \in I\right\}</math> is called the tail of <math>(I, \leq)</math> starting at <math>i.</math> The family <math>\operatorname{Tails}\left(x_{\bull}\right) := \left\{x_{\geq i} : i \in I\right\}</math> of all tails is a directed set with respect to <math>\,\supseteq;\,</math> in fact, it is even a prefilter.

====Neighborhoods====

If <math>T</math> is a [[topological space]] and <math>x_0</math> is a point in <math>T,</math> the set of all [[Topological neighbourhood|neighbourhoods]] of <math>x_0</math> can be turned into a directed set by writing <math>U \leq V</math> if and only if <math>U</math> contains <math>V.</math> For every <math>U,</math> <math>V,</math> and <math>W</math>{{hairsp}}:
* <math>U \leq U</math> since <math>U</math> contains itself.
* if <math>U \leq V</math> and <math>V \leq W,</math> then <math>U \supseteq V</math> and <math>V \supseteq W,</math> which implies <math>U \supseteq W.</math> Thus <math>U \leq W.</math>
* because <math>x_0 \in U \cap V,</math> and since both <math>U \supseteq U \cap V</math> and <math>V \supseteq U \cap V,</math> we have <math>U \leq U \cap V</math> and <math>V \leq U \cap V.</math>

====Finite subsets====

The set <math>\operatorname{Finite}(I)</math> of all finite subsets of a set <math>I</math> is directed with respect to <math>\,\subseteq\,</math> since given any two <math>A, B \in \operatorname{Finite}(I),</math> their union <math>A \cup B \in \operatorname{Finite}(I)</math> is an upper bound of <math>A</math> and <math>B</math> in <math>\operatorname{Finite}(I).</math> This particular directed set is used to define the sum <math>{\textstyle\sum\limits_{i \in I}} r_i</math> of a [[Generalized series (mathematics)|generalized series]] of an <math>I</math>-indexed collection of numbers <math>\left(r_i\right)_{i \in I}</math> (or more generally, the sum of [[Series (mathematics)#Abelian topological groups|elements in an]] [[abelian topological group]], such as [[Series (mathematics)#Series in topological vector spaces|vectors]] in a [[topological vector space]]) as the [[Limit of a net|limit of the net]] of [[partial sum]]s <math>F \in \operatorname{Finite}(I) \mapsto {\textstyle\sum\limits_{i \in F}} r_i;</math> that is:
<math display=block>\sum_{i \in I} r_i ~:=~ \lim_{F \in \operatorname{Finite}(I)} \ \sum_{i \in F} r_i ~=~ \lim \left\{\sum_{i \in F} r_i \,: F \subseteq I, F \text{ finite }\right\}.</math>

===Logic===

{{See also|Preorder#Preorders and partial orders on partitions}}

Let <math>S</math> be a [[Theory (mathematical logic)|formal theory]], which is a set of [[Sentence (mathematical logic)|sentences]] with certain properties (details of which can be found in [[Theory (mathematical logic)|the article on the subject]]). For instance, <math>S</math> could be a [[first-order theory]] (like [[Zermelo–Fraenkel set theory]]) or a simpler [[Propositional calculus|zeroth-order theory]]. The preordered set <math>(S, \Leftarrow)</math> is a directed set because if <math>A, B \in S</math> and if <math>C := A \wedge B</math> denotes the sentence formed by [[logical conjunction]] <math>\,\wedge,\,</math> then <math>A \Leftarrow C</math> and <math>B \Leftarrow C</math> where <math>C \in S.</math> 
If <math>S / \sim</math> is the [[Lindenbaum–Tarski algebra]] associated with <math>S</math> then <math>\left(S / \sim, \Leftarrow\right)</math> is a partially ordered set that is also a directed set.