subbasis

Let $(X,𝒯)$ be a topological space. A subset $𝒜\subseteq 𝒯$ is said to be a subbasis if the collection $ℬ$ of intersections of finitely many elements of $𝒜$ is a basis (http://planetmath.org/BasisTopologicalSpace) for $𝒯$.

Conversely, given an arbitrary collection $𝒜$ of subsets of $X$, a topology can be formed by first taking the collection $ℬ$ of finite intersections of members of $𝒜$ and then taking the topology $𝒯$ generated by $ℬ$ as basis. $𝒯$ will then be the smallest topology such that $𝒜\subseteq 𝒯$.