alternative characterization of ultrafilter

Let $X$ be a set. A filter $ℱ$ over $X$ is an ultrafilter if and only if it satisfies the following condition: if $A\coprod B=X$ (see disjoint union), then either $A\in ℱ$ or $B\in ℱ$.

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

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

1. A filter $ℱ$ over $X$ is an ultrafilter if and only if, whenever $A_1,\mathrm{\dots },A_n$ are subsets of $X$ such that ${\coprod }_{i=1}^n{A}_i=X$ then there exists exactly one $i$ such that $A_i\in ℱ$.

2. A filter $ℱ$ over $X$ is an ultrafilter if and only if, whenever $A_1,\mathrm{\dots },A_n$ are subsets of $X$ such that ${\bigcup }_{i=1}^n{A}_i=X$ then there exists an $i$ such that $A_i\in ℱ$.

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