Definition:Open Set Axioms

From ProofWiki
Jump to: navigation, search

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


Sources