Inmathematics, adirected set (or adirected preorder or afiltered set) is apreordered set in which every finite subset has anupper bound.^[1] In other words, it is a non-empty preordered set${\displaystyle A}$ such that for any${\displaystyle a}$ and${\displaystyle b}$ in${\displaystyle A}$ there exists${\displaystyle c}$ in${\displaystyle A}$ with${\displaystyle a\le c}$ and${\displaystyle b\le c}$.^[a] A directed set's preorder is called adirection.

The notion defined above is sometimes called anupward directed set. Adownward directed set is defined symmetrically,^[2] meaning that every finite subset has alower bound.^[3] 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.^[4]

Directed sets are a generalization of nonemptytotally ordered sets. That is, all totally ordered sets are directed sets (contrastpartially ordered sets, which need not be directed).Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise,lattices are directed sets both upward and downward.

Intopology, directed sets are used to definenets, which generalizesequences and unite the various notions oflimit used inanalysis. Directed sets also give rise todirect limits inabstract algebra and (more generally)category theory.

The set ofnatural numbers${\displaystyle \mathbb{N}}$ with the ordinary order${\displaystyle \le }$ is one of the most important examples of a directed set. Everytotally ordered set is a directed set, including${\displaystyle (\mathbb{N},\le ),}$${\displaystyle (\mathbb{N},\ge ),}$${\displaystyle (\mathbb{R},\le ),}$ and${\displaystyle (\mathbb{R},\ge ).}$

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

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

If${\displaystyle x_0}$ is areal number then the set${\displaystyle I:=\mathbb{R}\mathrm{\setminus }\{x_0\}}$ can be turned into a directed set by defining${\displaystyle a{\le }_{I}b}$ if${\displaystyle \left|a-x_0\right|\ge \left|b-x_0\right|}$ (so "greater" elements are closer to${\displaystyle x_0}$). We then say that the reals have beendirected towards${\displaystyle x_0.}$ This is an example of a directed set that isneitherpartially ordered nortotally ordered. This is becauseantisymmetry breaks down for every pair${\displaystyle a}$ and${\displaystyle b}$ equidistant from${\displaystyle x_0,}$ where${\displaystyle a}$ and${\displaystyle b}$ are on opposite sides of${\displaystyle x_0.}$ Explicitly, this happens when${\displaystyle \{a,b\}=\left\{x_0-r,x_0+r\right\}}$ for some real${\displaystyle r\ne 0,}$ in which case${\displaystyle a{\le }_{I}b}$ and${\displaystyle b{\le }_{I}a}$ even though${\displaystyle a\ne b.}$ Had this preorder been defined on${\displaystyle \mathbb{R}}$ instead of${\displaystyle \mathbb{R}\mathrm{\setminus }\{x_0\}}$ then it would still form a directed set but it would now have a (unique)greatest element, specifically${\displaystyle x_0}$; however, it still wouldn't be partially ordered. This example can be generalized to ametric space${\displaystyle (X,d)}$ by defining on${\displaystyle X}$ or${\displaystyle X\setminus \left\{x_0\right\}}$ the preorder${\displaystyle a\le b}$ if and only if${\displaystyle d\left(a,x_0\right)\ge d\left(b,x_0\right).}$

An element${\displaystyle m}$ of a preordered set${\displaystyle (I,\le )}$ is amaximal element if for every${\displaystyle j\in I,}$${\displaystyle m\le j}$ implies${\displaystyle j\le m.}$^[b]It is agreatest element if for every${\displaystyle j\in I,}$${\displaystyle j\le m.}$

Any preordered set with a greatest element is a directed set with the same preorder. For instance, in aposet${\displaystyle P,}$ everylower closure of an element; that is, every subset of the form${\displaystyle \{a\in P:a\le x\}}$ where${\displaystyle x}$ is a fixed element from${\displaystyle P,}$ 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.

Thesubset inclusion relation${\displaystyle \subseteq ,}$ along with itsdual${\displaystyle \supseteq ,}$ definepartial orders on any givenfamily of sets. A non-emptyfamily of sets is a directed set with respect to the partial order${\displaystyle \supseteq }$ (respectively,${\displaystyle \subseteq }$) 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${\displaystyle I}$ of sets is directed with respect to${\displaystyle \supseteq }$ (respectively,${\displaystyle \subseteq }$) if and only if

for all${\displaystyle A,B\in I,}$ there exists some${\displaystyle C\in I}$ such that${\displaystyle A\supseteq C}$ and${\displaystyle B\supseteq C}$ (respectively,${\displaystyle A\subseteq C}$ and${\displaystyle B\subseteq C}$)

or equivalently,

for all${\displaystyle A,B\in I,}$ there exists some${\displaystyle C\in I}$ such that${\displaystyle A\cap B\supseteq C}$ (respectively,${\displaystyle A\cup B\subseteq C}$).

Many important examples of directed sets can be defined using these partial orders. For example, by definition, aprefilter orfilter base is a non-emptyfamily of sets that is a directed set with respect to thepartial order${\displaystyle \supseteq }$ and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would then be agreatest element with respect to${\displaystyle \supseteq }$). Everyπ-system, which is a non-emptyfamily of sets that is closed under the intersection of any two of its members, is a directed set with respect to${\displaystyle \supseteq .}$ Everyλ-system is a directed set with respect to${\displaystyle \subseteq .}$ Everyfilter,topology, andσ-algebra is a directed set with respect to both${\displaystyle \supseteq }$ and${\displaystyle \subseteq .}$

By definition, anet is a function from a directed set and asequence is a function from the natural numbers${\displaystyle \mathbb{N}.}$ Every sequence canonically becomes a net by endowing${\displaystyle \mathbb{N}}$ with${\displaystyle \le .}$

If${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ is anynet from a directed set${\displaystyle (I,\le )}$ then for any index${\displaystyle i\in I,}$ the set${\displaystyle x_{\ge i}:=\left\{x_j:j\ge i\text{ with }j\in I\right\}}$ is called the tail of${\displaystyle (I,\le )}$ starting at${\displaystyle i.}$ The family${\displaystyle \mathrm{Tails}\left(x_{\bullet }\right):=\left\{x_{\ge i}:i\in I\right\}}$ of all tails is a directed set with respect to${\displaystyle \supseteq ;}$ in fact, it is even a prefilter.

If${\displaystyle T}$ is atopological space and${\displaystyle x_0}$ is a point in${\displaystyle T,}$ the set of allneighbourhoods of${\displaystyle x_0}$ can be turned into a directed set by writing${\displaystyle U\le V}$ if and only if${\displaystyle U}$ contains${\displaystyle V.}$ For every${\displaystyle U,}$${\displaystyle V,}$ and${\displaystyle W}$ :

• ${\displaystyle U\le U}$ since${\displaystyle U}$ contains itself.
• if${\displaystyle U\le V}$ and${\displaystyle V\le W,}$ then${\displaystyle U\supseteq V}$ and${\displaystyle V\supseteq W,}$ which implies${\displaystyle U\supseteq W.}$ Thus${\displaystyle U\le W.}$
• because${\displaystyle x_0\in U\cap V,}$ and since both${\displaystyle U\supseteq U\cap V}$ and${\displaystyle V\supseteq U\cap V,}$ we have${\displaystyle U\le U\cap V}$ and${\displaystyle V\le U\cap V.}$

The set${\displaystyle \mathrm{Finite}(I)}$ of all finite subsets of a set${\displaystyle I}$ is directed with respect to${\displaystyle \subseteq }$ since given any two${\displaystyle A,B\in \mathrm{Finite}(I),}$ their union${\displaystyle A\cup B\in \mathrm{Finite}(I)}$ is an upper bound of${\displaystyle A}$ and${\displaystyle B}$ in${\displaystyle \mathrm{Finite}(I).}$ This particular directed set is used to define the sum${\displaystyle \sum _{i\in I}{r}_i}$ of ageneralized series of an${\displaystyle I}$-indexed collection of numbers${\displaystyle {\left(r_i\right)}_{i\in I}}$ (or more generally, the sum ofelements in anabelian topological group, such asvectors in atopological vector space) as thelimit of the net ofpartial sums${\displaystyle F\in \mathrm{Finite}(I)\mapsto \sum _{i\in F}{r}_i;}$ that is:$${\displaystyle \sum _{i\in I}{r}_i:=\underset{F\in \mathrm{Finite}(I)}{lim}\sum _{i\in F}{r}_i=lim\left\{\sum _{i\in F}{r}_i:F\subseteq I,F\text{ finite }\right\}.}$$

Let${\displaystyle S}$ be aformal theory, which is a set ofsentences with certain properties (details of which can be found inthe article on the subject). For instance,${\displaystyle S}$ could be afirst-order theory (likeZermelo–Fraenkel set theory) or a simplerzeroth-order theory. The preordered set${\displaystyle (S,\Leftarrow )}$ is a directed set because if${\displaystyle A,B\in S}$ and if${\displaystyle C:=A\wedge B}$ denotes the sentence formed bylogical conjunction${\displaystyle \wedge ,}$ then${\displaystyle A\Leftarrow C}$ and${\displaystyle B\Leftarrow C}$ where${\displaystyle C\in S.}$ If${\displaystyle S/\sim }$ is theLindenbaum–Tarski algebra associated with${\displaystyle S}$ then${\displaystyle \left(S/\sim ,\Leftarrow \right)}$ is a partially ordered set that is also a directed set.

Directed set is a more general concept than (join) semilattice: everyjoin semilattice is a directed set, as the join or least upper bound of two elements is the desired${\displaystyle c.}$ The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111}ordered bitwise (e.g.${\displaystyle 1000\le 1011}$ holds, but${\displaystyle 0001\le 1000}$ does not, since in the last bit 1 > 0), where {1000,0001} has three upper bounds but noleast upper bound, cf. picture. (Also note that without 1111, the set is not directed.)

The order relation in a directed set is not required to beantisymmetric, and therefore directed sets are not alwayspartial orders. However, the termdirected set is also used frequently in the context of posets. In this setting, a subset${\displaystyle A}$ of a partially ordered set${\displaystyle (P,\le )}$ is called adirected subset if it is a directed set according to the same partial order: in other words, it is not theempty set, and every pair of elements has an upper bound. Here the order relation on the elements of${\displaystyle A}$ is inherited from${\displaystyle P}$; for this reason, reflexivity and transitivity need not be required explicitly.

A directed subset of a poset is not required to bedownward closed; a subset of a poset is directed if and only if its downward closure is anideal. 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 afilter.

Directed subsets are used indomain theory, which studiesdirected-complete partial orders.^[5] These are posets in which every upward-directed set is required to have aleast upper bound. In this context, directed subsets again provide a generalization of convergent sequences.^{[further explanation needed]}

• Centered set – Order theory
• Filtered category
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results
• Linked set – Mathematical concept regarding posets in (partial) order theory
• Net (mathematics) – Generalization of a sequence of points

• Kelley, John L. (1975) [1955].General Topology.Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag.ISBN 978-0-387-90125-1.OCLC 1365153.
• Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S. (2003).Continuous Lattices and Domains. Cambridge University Press.doi:10.1017/CBO9780511542725.ISBN 9780511542725.OCLC 7334257218.

• Binary relations
• General topology
• Order theory

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