Definition:Filter on Set/Definition 1

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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:

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

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