Definition 5.2.1 (Filter).label Let $X$ be a set, a filter $\fF \subset 2^{X}$ is a non-empty family of sets such that:

1. (F1)

   For any $E \in \fF$ and $X \supset F \supset E$, $F \in \fF$.

2. (F2)

   For any $E, F \in \fF$, $E \cap F \in \fF$.

3. (F3)

   $\emptyset \not\in \fF$