Directed set

(that is, a preorder), with the additional property that every pair of elements has an upper bound.

The notion defined above is sometimes called an upward directed set.

A downward directed set is defined analogously,[2] meaning that every pair of elements is bounded below.

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.

In this definition, the existence of an upper bound of the empty subset implies that

" but in which the ordering rule only applies to pairs of elements on the same side of

of pairs of natural numbers can be made into a directed set by defining

then it would still form a directed set but it would now have a (unique) greatest element, specifically

every lower closure of an element; that is, every subset of the form

define partial orders on any given family of sets.

) 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.

) if and only if or equivalently, Many important examples of directed sets can be defined using these partial orders.

For example, by definition, a prefilter or filter base 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 filter, topology, and σ-algebra is a directed set with respect to both

(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

be a formal theory, which is a set of sentences with certain properties (details of which can be found in the article on the subject).

denotes the sentence formed by logical conjunction

The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} ordered bitwise (e.g.

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 directed set is also used frequently in the context of posets.

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.

; 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.

[5] 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.