Directed set

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. Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets (but not all partially ordered sets). In topology, directed sets are used to... more

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