# Axiom:Open Set Axioms

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## Definition

Let $S$ be a set.

The **open set axioms** are the conditions under which elements of a subset $\tau \subseteq \powerset S$ of the power set of $S$ need to satisfy in order to be open sets of the topology $\tau$ on $S$:

\((\text O 1)\) | $:$ | The union of an arbitrary subset of $\tau$ is an element of $\tau$. | |||||||

\((\text O 2)\) | $:$ | The intersection of any two elements of $\tau$ is an element of $\tau$. | |||||||

\((\text O 3)\) | $:$ | $S$ is an element of $\tau$. |

## Also see

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

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets: Definition $1$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction