# Definition:Ultrafilter on Set/Definition 4

## Definition

Let $S$ be a non-empty set.

Let $\FF$ be a non-empty set of subsets of $S$.

Then $\FF$ is an ultrafilter on $S$ if and only if both of the following hold:

$\FF$ has the finite intersection property
For all $U \subseteq S$, either $U \in \FF$ or $U^\complement \in \FF$

where $U^\complement$ is the complement of $U$ in $S$.

## Sources

• 2005: R.E. HodelRestricted versions of the Tukey-Teichmuller Theorem that are equivalent to the Boolean prime ideal theorem (Arch. Math. Logic Vol. 44: pp. 459 – 472)