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Directed set

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

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 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.