Subbasis

Definition

A subbasis $\mathcal{S}$ for a topology $\mathcal{T}$ on $X$ is a collection of subsets of $X$ whose union equals $X$. $\mathcal{S}\subset 2^X\text{s.t.}\bigcup \mathcal{S}=X$

Or, Equivalently, a subset $\mathcal{S}$ of $\mathcal{T}$ is a subbasis for $\mathcal{T}$ if the family of all finite intersections of members of the collection $\{{\bigcap }_{u=1}^n{S}_i|S_i\in \mathcal{S},n\in \mathbb{N}\}$ is a basis for $\mathcal{T}$.

Basis Generated By a Subbasis

$\mathcal{B}=\left\{{\bigcap }_{i=1}^n{S}_i|S_i\in \mathcal{S}\right\}$

A basis can be defined as a set of all finite intersections of a subbasis.

Examples

A Standard Topology on $\mathbb{R}$ can be generated by a subbasis $\mathcal{S}=\{(a,\infty ),(-\infty ,b)\subset \mathbb{R}|a,b\in \mathbb{R}\}$. Every basis element $B\in \mathcal{B}=\{(a,b)\subset \mathbb{R}|a<b\}$ can be generated by the subbasis $\mathcal{S}$.