Definition:Ultrafilter on Set

Let $$X$$ be a set and $$\mathcal{F} \subset \mathcal{P}(X)$$ be a filter on $$X$$. $$\mathcal{F}$$ is called an ultrafilter (on $$X$$) if it is not contained in a larger filter (on $$X$$), i.e. if the following condition holds: If $$\mathcal{G}$$ is a filter on $$X$$ and $$\mathcal{F} \subseteq \mathcal{G}$$ holds then $$\mathcal{F} = \mathcal{G}$$.