In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set  together with a reflexive and transitive binary relation  (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any  and  in  there must exist  in  with  and  A directed set's preorder is called a direction.

The notion defined above is sometimes called an . A  is defined analogously, meaning that every pair of elements is bounded below. 
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.

Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast  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.

In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

Equivalent definition

In addition to the definition above, there is an equivalent definition. A directed set is a set  with a preorder such that every finite subset of  has an upper bound. In this definition, the existence of an upper bound of the empty subset implies that  is nonempty.

Examples

The set of natural numbers  with the ordinary order  is one of the most important examples of a directed set (and so is every totally ordered set). By definition, a  is a function from a directed set and a sequence is a function from the natural numbers  Every sequence canonically becomes a net by endowing  with  

If  is a real number then the set  can be turned into a directed set by defining  if  (so "greater" elements are closer to ). We then say that the reals have been directed towards  This is an example of a directed set that is  partially ordered nor totally ordered. This is because antisymmetry breaks down for every pair  and  equidistant from  where  and  are on opposite sides of  Explicitly, this happens when  for some real  in which case  and  even though  Had this preorder been defined on  instead of  then it would still form a directed set but it would now have a (unique) greatest element, specifically ; however, it still wouldn't be partially ordered. This example can be generalized to a metric space  by defining on  or  the preorder  if and only if 

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

Maximal and greatest elements

An element  of a preordered set  is a maximal element if for every   implies 
It is a greatest element if for every  

Any preordered set with a greatest element is a directed set with the same preorder. 
For instance, in a poset  every lower closure of an element; that is, every subset of the form  where  is a fixed element from  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.

Product of directed sets

Let  and  be directed sets. Then the Cartesian product set  can be made into a directed set by defining  if and only if  and  In analogy to the product order this is the product direction on the Cartesian product.  For example, the set  of pairs of natural numbers can be made into a directed set by defining  if and only if  and

Subset inclusion

The subset inclusion relation  along with its dual  define partial orders on any given family of sets. 
A non-empty family of sets is a directed set with respect to the partial order  (respectively, ) 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  of sets is directed with respect to  (respectively, ) if and only if 
for all  there exists some  such that  and  (respectively,  and ) 
or equivalently, 
for all  there exists some  such that  (respectively, ).

Many important examples of directed sets can be defined using these partial orders. 
For example, by definition, a  or  is a non-empty family of sets that is a directed set with respect to the partial order  and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would then be a greatest element with respect to ). 
Every -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  Every λ-system is a directed set with respect to  Every filter, topology, and σ-algebra is a directed set with respect to both  and  
If  is any net from a directed set  then for any index  the set  is called the tail of  starting at  The family  of all tails is a directed set with respect to  in fact, it is even a prefilter.

If  is a topological space and  is a point in  set of all neighbourhoods of  can be turned into a directed set by writing  if and only if  contains  For every   and :
  since  contains itself.
 if  and  then  and  which implies  Thus 
 because  and since both  and  we have  and 

The set  of all finite subsets of a set  is directed with respect to  since given any two  their union  is an upper bound of  and  in  This particular directed set is used to define the sum  of a generalized series of an -indexed collection of numbers  (or more generally, the sum of elements in an abelian topological group, such as vectors in a topological vector space) as the limit of the net of partial sums  that is:

Contrast with semilattices

Directed set is a more general concept than (join) semilattice: every join semilattice is a directed set, as the join or least upper bound of two elements is the desired   The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} ordered bitwise (e.g.  holds, but  does not, since in the last bit 1 > 0), where {1000,0001} has three upper bounds but no  upper bound, cf. picture. (Also note that without 1111, the set is not directed.)

Directed subsets
The order relation in a directed set is not required to be antisymmetric, and therefore directed sets are not always partial orders. However, the term  is also used frequently in the context of posets. In this setting, a subset  of a partially ordered set  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  is inherited from ; for this reason, reflexivity and transitivity need not be required explicitly.

A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if its downward closure is an 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.

Directed subsets are used in domain theory, which studies directed-complete partial orders. 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.

See also

Notes

References

 J. L. Kelley (1955), General Topology.
 Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. .

Binary relations
General topology
Order theory