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 $A$.

Also see

 * Basis (Topology)
 * Filter Sub-Basis

Linguistic Variance
Some sources omit the hyphen and write subbasis.

The term sub-base (or subbase) is also seen sometimes.