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
Famous quotes containing the words directed and/or set:
“Life is like a box of chocolates. You never know what you’re gonna get.”
—Eric Roth, U.S. screenwriter. Directed by Robert Zemekis. Forrest Gump (Tom Hanks)
“He could walk, or rather turn about in his little garden, and feel more solid happiness from the flourishing of a cabbage or the growing of a turnip than was ever received from the most ostentatious show the vanity of man could possibly invent. He could delight himself with thinking, “Here will I set such a root, because my Camilla likes it; here, such another, because it is my little David’s favorite.””
—Sarah Fielding (1710–1768)