Definition 4.1 (Filter). Let $J$ be a set. A filter $F$ on $J$ is a non-empty subset of $P(J)$ such that:

• $\varnothing \notin F$.

• $\forall A,B\in F,A\cap \in F$ (“closed under finite intersections”).

• $\forall A\in F$, if $A\subseteq B\subseteq B\subseteq J$, then $B\in F$ (“closed under super set”).