Directedness

Directedness is a property of a subset of a partially ordered set that ensures any two of its elements can be extended to a common future within the subset. Formally, if (P, ≤) is a poset and D ⊆ P is nonempty, D is directed if for every a,b ∈ D there exists c ∈ D with a ≤ c and b ≤ c. A directed set is the pair (D, ≤) restricted to such a subset; the term is widely used in order theory, topology, and category theory.

Special cases and examples: In a total order, any nonempty subset is directed, because for any a,b there is one of a or b that is larger and still lies in the subset. In general posets, directedness does not require upward closure or containing all upper bounds in P, only that D itself contains a common upper bound for every pair of its elements.

Allies and uses: Directedness is central to the theory of nets, where convergence is defined with respect to directed index sets, generalizing sequences. It also appears in the construction of colimits and filtered colimits in category theory, and in inverse systems (projective limits) where directed sets index objects with connecting morphisms.

Notes: The dual notion is filteredness; the term “directed” emphasizes the existence of common upper bounds, whereas filteredness emphasizes common lower bounds in the opposite order. See also directed set, net (topology), inverse system, colimit.