Directed set Guide, Meaning , Facts, Information and Description

In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties:

• 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 antisymmetric 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 iff

• A is not the empty set,
• 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 theory, 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 relation, partial order, semilattice.

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