Definition:Sub-Basis

Analytic Sub-Basis
Let $$T = \left\{{A, \vartheta}\right\}$$ be a topological space.

Let $$\mathcal{S} \subseteq \vartheta$$ be such that every $$U \in \vartheta$$ is a union of finite intersections of sets from $$\mathcal{S}$$.

Then $$\mathcal{S}$$ is a a(n) (analytic) sub-basis for $$\vartheta$$.

Synthetic Sub-Basis
Let $$A$$ be a set.

Let $$\mathcal{S} \subseteq \mathcal{P} \left({A}\right)$$, where $$\mathcal{P} \left({A}\right)$$ is the power set of $$A$$.

The collection of all finite intersections of sets from $$\left\{{A}\right\} \cup \mathcal{S}$$ forms a synthetic basis for $$A$$.

This is proved in Synthetic Basis Formed from Synthetic Sub-Basis.

Then $$\mathcal{S}$$ is a (synthetic) sub-basis for $$A$$.

Note that by this construction, any collection of subsets of $$A$$ can form a synthetic basis and thus a topology on $$A$$.