Axiom:Filter on Set Axioms

Definition

Let $S$ be a set.

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

Let $\FF \subset \powerset S$.

$\FF$ is said to satisfy the filter on set axioms if and only if:

$(\text F 1)$	$:$	$\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 $

$\FF$ is said to satisfy the filter on set axioms if and only if:

$(\text F 1)$	$:$	$\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 $

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

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