Definition:Open Set Axioms

Definition

Let $S$ be a set.

The open set axioms are the conditions under which elements of a subset $\tau \subseteq \powerset S$ 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 $\tau$.

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.
Definition:Closed Set Axioms

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

\((\text O 1)\)	$:$	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 $\tau$.