I have come across to the following question :
Let $\mathscr{T}_\alpha$ be a family of topologies on $ X$ . Show that there is a unique smallest topology on $X$ containing all the collections $\mathscr{T}_\alpha$ , and a unique largest topology contained in all $\mathscr{T}_\alpha$.
I think that the unique smallest topology equals the union of all the \mathscr{T}_\alpha's, and the unique largest topology contained in all $\mathscr{T}_\alpha$ equals the intersection of all the \mathscr{T}_\alpha's .