Axiom:Filter on Set Axioms/Axioms 1

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Definition

Let $S$ be a set.

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

Let $\FF \subset \powerset S$.


$\FF$ is a filter on $S$ if and only if $\FF$ satisfies the following axioms:

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

These criteria are called the filter on set axioms.


Also see


Sources