# 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 $\tau$. |

## Also see

- Empty Set is Element of Topology, which demonstrates that it is not necessary to specify that $\varnothing \in \tau$ as this follows directly from the axioms.
- Definition:Closed Set Axioms

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): $\S 1.1$ - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$