Question

Prove that any collection of subsets of a set X is a sub base for some topology on X

Answer 1

Proof: Let $\mathscr{B}$ be the family of finite intersections of members of $\mathcal{S}$ . Suppose first that S is a sub-base for $\mathscr{I}$ . We want to show that $\mathscr{I}$ is the smallest topology on X containing $\mathcal{S}$ . Now since $\mathcal{S} \subset \mathscr{B}$ and $\mathscr{B} \subset \mathscr{I}$ we at least have that $\mathscr{I}$ contains S. Suppose $\mathcal{U}$ is some other topology on X such that $\mathcal{S} \subset \mathcal{U}$ . We have to show that $\mathscr{I} \subset \mathcal{U}$ . Now since $\mathcal{U}$ is closed under finite intersections and $S \subset \mathcal{U}$ , $\mathcal{U}$ contains all finite intersections of members of $\mathcal{S}$ ,

i.e. $\mathscr{B} \subset \mathcal{U}$ . But again since $\mathcal{U}$ is closed under arbitrary unions and each member of $\mathscr{I}$ can be written as union of some members of $\mathscr{B}$ (by definition of a base), it follows that $\mathscr{I} \subset \mathcal{U}$ .