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): $\text{I}: \ \S 1$