# Definition:Closed Set Axioms

Let $S$ be a set.
The closed set axioms are the conditions under which a subset $F \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ consists of the closed sets of a topology on $S$:
 $(C1)$ $:$ The intersection of an arbitrary subset of $F$ is an element of $F$. $(C2)$ $:$ The union of any two elements of $F$ is an element of $F$. $(C3)$ $:$ $\varnothing$ is an element of $F$.