Let $T_a$ be a family of topologies on a set $X$. What is the smallest topology containing all the $T_a$? Obviously, the smallest it could be is the union of all the $T_a$, but that's not always a topology. So is it just the topology generated by the subbasis $\bigcup T_a$? I feel like this is too simple of an answer, since the question is phrased asking to prove that there is a unique smallest topology containing all the $T_a$.
(I'm working through Munkres' Topology on my own.)
Best Answer
Yes, it's the topology that has $\bigcup T_a$ as a subbasis. It is also the final topology with respect to the family $\iota_a\colon (X,T_a) \to X$, where $\iota_a$ is the identity map for all $a$.
It is that simple, though you need to say a word or two about the fact that it is indeed coarser than any topology containing all $T_a$ (and hence it's uniquely determined as the smallest element in a partially ordered set; not only a minimal element, which need not be unique). But a few words really suffice.