In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.
Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets (contrast partially ordered sets which need not be directed). In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.
Read more about Directed Set: Equivalent Definition, Examples, Contrast With Semilattices, Directed Subsets
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