Home > Directed set

• a ≤ a for all a in A ( reflexivity)
• if a ≤ b and b ≤ c, then a ≤ c ( transitivity)
• for any two a and b in A, there exists a c in A with a ≤ c and b ≤ c (directedness)

Directed sets in this form are used to define nets in topology. Nets generalize sequences and unite the various notions of limit used in analysis.

Examples of directed sets include:

• The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).
• If x₀ is a real number, we can turn the set R - {x₀} into a directed set by writing a ≤ b if and only if
  |a - x₀| ≥ |b - x₀|. We then say that the reals have been directed towards x₀. This is not a partial order.
• If T is a topological space and x₀ is a point in T, we turn the set of all neighbourhoods of x₀ into a directed set by writing U ≤ V if and only if U contains V.
  • For every U: U ≤ U; since U contains itself.
  • For every U,V,W: if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then U contains W.
  • For every U, V: there exists the set U ∩V such that U ≤ U ∩V and V ≤ U ∩V; since both U and V contain U ∩V.
• In a poset P, every subset of the form {a| a in P, a ≤x}, where x is a fixed element from P, is directed.

Note that directed sets need not be antisymmetricIn mathematics, a binary relation R over a set X is antisymmetric if it holds for all a and b in X that if a is related to b and b is related to a then a b''. In notation, this is: : Strict inequality is antisymmetric; since a < b and b < a is impossible, and therefore in general are not partial orders. However, the term is also frequently used in the context of posets. In this setting, a subset A of a partially ordered set (P,≤) is called a directed subset iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P

• A is not the empty setAbstract algebra Algebra Set theory In mathematics, the empty set is the set with no elements. Notation The standard notation for denoting the empty set, invented by Nicholas Bourbaki, is the symbol , also written as or ∅, and sometimes approximated,
• for any two a and b in A, there exists a c in A with a ≤ c and b ≤ c (directedness),

where the order of the elements of A is inherited from P. For this reason, reflexivity and transitivity need not be required explicitly.

Directed subsets are most commonly used in domain theoryDomain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains . Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where, where one studies orders for which these sets are required to have a least upper bound. Thus, directed subsets provide a generalization of (converging) sequences in the setting of partial orders as well.

Compare: equivalence relationIn mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive symmetric and transitive i. if the relation is written as ~ it holds for all a b and c in X that # (Reflexivity) a ~ a # (Symmetry) if a ~ b then b ~ a # (Trans, partial orderIn mathematics, partially ordered sets or posets for short, are special binary relations which formalize the intuitive concept of an ordering. Partially ordered sets are studied in order theory and a much more detailed introduction to the field can be fou, semilattice.