Union from Synthetic Basis is Topology/Proof 2

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

Let $\displaystyle \tau = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$.

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

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

Proof
We use Equivalent Definitions of Topology Generated by Synthetic Basis.

Also see

 * Topology Generated by Synthetic Basis
 * Generated Topology