Definition:Topology Generated by Synthetic Sub-Basis/Definition 1

Definition
Let $X$ be a set.

Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a synthetic sub-basis on $X$.

Let $\mathcal B$ be the synthetic basis on $X$ formed from the synthetic sub-basis $\mathcal S$.

The topology generated by $\mathcal S$, denoted $\tau \left({\mathcal S}\right)$, is defined as the topology on $X$ generated by the synthetic basis $\mathcal B$.

Equivalence of Definitions
This definition is shown to be equivalent to Generated Topology: Definition 2 in Equivalence of Definitions of Generated Topology.

Also see

 * Existence and Uniqueness of Generated Topology
 * Initial Topology
 * Basis
 * Topology Generated by Synthetic Basis