Definition:Filter on Set/Definition 2

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Let $S$ be a set.

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

A filter on $S$ (or filter of $S$) is a set $\FF \subset \powerset S$ which satisfies the following conditions:

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

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