alternative characterization of ultrafilter


Let X be a set. A filter over X is an ultrafilterMathworldPlanetmathPlanetmath if and only if it satisfies the following condition: if AB=X (see disjoint unionMathworldPlanetmath), then either A or B.

This result can be generalized somewhat: a filter over X is an ultrafilter if and only if it satisfies the following condition: if AB=X (see union), then either A or B.

This theorem can be extended to the following two propositionsPlanetmathPlanetmath about finite unions:

  1. 1.

    A filter over X is an ultrafilter if and only if, whenever A1,,An are subsets of X such that i=1nAi=X then there exists exactly one i such that Ai.

  2. 2.

    A filter over X is an ultrafilter if and only if, whenever A1,,An are subsets of X such that i=1nAi=X then there exists an i such that Ai.

Title alternative characterization of ultrafilter
Canonical name AlternativeCharacterizationOfUltrafilter
Date of creation 2013-03-22 14:42:20
Last modified on 2013-03-22 14:42:20
Owner yark (2760)
Last modified by yark (2760)
Numerical id 13
Author yark (2760)
Entry type Theorem
Classification msc 54A20