subbasisSubbasis2002-01-06 15:50:302004-06-22 17:10:17Definitiontopology\usepackage{amssymb}
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\usepackage{xypic}Let $(X,\mathcal{T})$ be a topological space. A subset $\mathcal{A}\subseteq\mathcal{T}$ is said to be a \emph{subbasis} if the collection $\mathcal{B}$ of intersections of finitely many elements of $\mathcal{A}$ is a \PMlinkname{basis}{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 $\mathcal{A}\subseteq\mathcal{T}$.