# Definition:Filter on Set/Definition 2

## Definition

Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

A set $\FF \subset \powerset S$ is a filter on $S$ (or filter of $S$) if and only if $\FF$ satisfies the filter on set axioms:

 $(\text F 1)$ $:$ $\ds S \in \FF$ $(\text F 2)$ $:$ $\ds \O \notin \FF$ $(\text F 3)$ $:$ $\ds \forall n \in \N: U_1, \ldots, U_n \in \FF \implies \bigcap_{i \mathop = 1}^n U_i \in \FF$ $(\text F 4)$ $:$ $\ds \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF$