directed set

A directed set is a partially ordered setMathworldPlanetmath (A,) such that whenever a,bA there is an xA such that ax and bx.

A subset BA is said to be residual if there is aA such that bB whenever ab, and cofinal if for each aA there is bB such that ab.

A directed set is sometimes called an upward-directed set. We may also define the dual notion: a downward-directed set (or filtered set) is a partially ordered set (A,) such that whenever a,bA there is an xA such that xa and xb.

Note: Many authors do not require to be antisymmetric, so that it is only a pre-order (rather than a partial orderMathworldPlanetmath) with the given property. Also, it is common to require A to be non-empty.

Title directed set
Canonical name DirectedSet
Date of creation 2013-03-22 12:54:00
Last modified on 2013-03-22 12:54:00
Owner yark (2760)
Last modified by yark (2760)
Numerical id 11
Author yark (2760)
Entry type Definition
Classification msc 06A06
Synonym upward-directed set
Synonym upward directed set
Related topic Cofinality
Related topic AccumulationPointsAndConvergentSubnets
Defines residual
Defines cofinal
Defines downward-directed set
Defines downward directed set
Defines filtered set