Editing Directed set (section)

===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>