Axiom:Closed Set Axioms

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Let $S$ be a set.

The closed set axioms are the conditions under which a subset $F \subseteq \powerset S$ of the power set of $S$ consists of the closed sets of a topology on $S$:

\((\text C 1)\)   $:$   The intersection of an arbitrary subset of $F$ is an element of $F$.      
\((\text C 2)\)   $:$   The union of any two elements of $F$ is an element of $F$.      
\((\text C 3)\)   $:$   $\O$ is an element of $F$.      

Also see