# Definition:Open Set Axioms

## Definition

Let $S$ be a set.

The open set axioms are the conditions under which elements of a subset $\tau \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ need to satisfy in order to be open sets of the topology $\tau$ on $S$:

 $(O1)$ $:$ The union of an arbitrary subset of $\tau$ is an element of $\tau$. $(O2)$ $:$ The intersection of any two elements of $\tau$ is an element of $\tau$. $(O3)$ $:$ $S$ is an element of $\tau$.