Definition:Topology Generated by Synthetic Sub-Basis

Definition 1
Let $X$ be a set.

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a subset of the power set of $X$.

Then $\mathcal G$ is a synthetic sub-basis on $X$.

Let $\mathcal B$ be the synthetic basis on $X$ generated by the synthetic sub-basis $\mathcal G$.

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

Definition 2
Let $X$ be a set.

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a subset of the power set of $X$.

The topology generated by $\mathcal G$, denoted $\tau \left({\mathcal G}\right)$, is defined as the unique topology on $X$ that satisfies the following axioms:
 * $\left({1}\right): \quad$ $\mathcal G \subseteq \tau \left({\mathcal G}\right)$.
 * $\left({2}\right): \quad$ For any topology $\mathcal T$ on $X$, the implication $\mathcal G \subseteq \mathcal T \implies \tau \left({\mathcal G}\right) \subseteq \mathcal T$ holds.

That is, $\tau \left({\mathcal G}\right)$ is the coarsest topology on $X$ for which every element of $\mathcal G$ is open.

Also see

 * Equivalence of Definitions of Generated Topology
 * Existence and Uniqueness of Generated Topology
 * Initial Topology
 * Basis