Definition:Filter on Set/Definition 1

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 $S$ 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 U, V \in \FF \implies U \cap V \in \FF $
$(\text F 4)$	$:$	$\ds \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF $

\((\text F 1)\)	$:$	\(\ds S \in \FF \)
\((\text F 2)\)	$:$	\(\ds \O \notin \FF \)
\((\text F 3)\)	$:$	\(\ds U, V \in \FF \implies U \cap V \in \FF \)
\((\text F 4)\)	$:$	\(\ds \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF \)