Filter

Idea

An ultrafilter is a way of making a consistent binary decision about which subsets of a set count as ‘large’.

Definition

Let $X$ be a set. A filter $\mathcal{F}$ on $X$ is a collection of subsets of $X$ such that:

• $X\in \mathcal{F}$.
• $\varnothing \notin \mathcal{F}$.
• If $A,B\in \mathcal{F}$, then $A\cap B\in \mathcal{F}$.
• If $A\in \mathcal{F}$ and $A\subseteq B\subseteq X$, then $B\in \mathcal{F}$.

An ultrafilter $\mathcal{U}$ is a maximal filter:

$$A\in \mathcal{U}\vee (X-A)\in \mathcal{U}$$

Remarks

The lemma that every proper filter on a set $X$ extends to an ultrafilter relies on choice for the proof.