Definition (Filter). A filter is a non-empty collection $F$ of subsets of $\mathbb{N}$ satisfying:

• (a) $\varnothing \notin F$.
• (b) If $A\in F$, $A\subset B$, then $B\in F$ (‘upset’).
• (c) If $A\in F$, $B\in F$ then $A\cap B\in F$ (closed under finite intersections).