Union from Synthetic Basis is Topology

Theorem
Let $\BB$ be a synthetic basis on a set $X$.

Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.

Then $\tau$ is a topology on $X$.

$\tau$ is called the topology arising from, or generated by, the basis $\BB$.

Also see

 * Definition:Topology Generated by Synthetic Basis
 * Definition:Topology Generated by Synthetic Sub-Basis