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 $X$. Define:
 * $\displaystyle \mathcal B = \left\{{\bigcap \mathcal F: \mathcal F \subseteq \mathcal S, \, \mathcal F \text{ is finite}}\right\}$

That is, $\mathcal B$ is the set of all finite intersections of sets in $\mathcal S$.

Note that $\mathcal F$ is allowed to be empty in the above definition.

The topology generated by $\mathcal S$, denoted $\tau \left({\mathcal S}\right)$, is defined as:
 * $\displaystyle \tau \left({\mathcal S}\right) = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$

It follows directly from Synthetic Basis formed from Synthetic Sub-Basis and Union from Synthetic Basis is Topology that $\tau \left({\mathcal S}\right)$ is a topology on $X$.

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