# alternative characterization of ultrafilter

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

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

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

1. 1.

A filter $\mathcal{F}$ over $X$ is an ultrafilter if and only if, whenever $A_{1},\dots,A_{n}$ are subsets of $X$ such that $\coprod_{i=1}^{n}A_{i}=X$ then there exists exactly one $i$ such that $A_{i}\in\mathcal{F}$.

2. 2.

A filter $\mathcal{F}$ over $X$ is an ultrafilter if and only if, whenever $A_{1},\dots,A_{n}$ are subsets of $X$ such that $\bigcup_{i=1}^{n}A_{i}=X$ then there exists an $i$ such that $A_{i}\in\mathcal{F}$.

Title alternative characterization of ultrafilter AlternativeCharacterizationOfUltrafilter 2013-03-22 14:42:20 2013-03-22 14:42:20 yark (2760) yark (2760) 13 yark (2760) Theorem msc 54A20