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In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation , with the additional property that every pair of elements has an upper bound.Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets. 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 category theory.