Definition:Topology Generated by Synthetic Sub-Basis/Definition 2

Definition
Let $X$ be a set.

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

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

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

Equivalence of Definitions
This definition is shown to be equivalent to Generated Topology: Definition 1 in Equivalence of Definitions of Generated Topology.

Also see

 * Existence and Uniqueness of Generated Topology, which proves the existence and uniqueness of $\tau \left({\mathcal S}\right)$ as defined here
 * Initial Topology
 * Basis
 * Topology Generated by Synthetic Basis