# subbasis

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

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

Title subbasis Subbasis 2013-03-22 12:10:32 2013-03-22 12:10:32 evin290 (5830) evin290 (5830) 10 evin290 (5830) Definition msc 54A99 subbasic subbasic subbase Basis BasisTopologicalSpace