how to find the unique smallest topology? 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 $\mathscr{T}_\alpha's$ .  
 A: Many texts discuss this question, perhaps leaving some of the verifications to the reader.  For instance you could look here.
(I guess that by writing these notes and posting them on the web I am implicitly giving the opinion that facts like these are too basic and foundational to make for good exercises for the student.  There is no lack of things to ask a student to solve, so in my opinion as an instructor or a writer you might as well write out the really basic proofs in full.  I note that, for instance, Bourbaki has this philosophy: a lot of "his" exercises are not only more challenging than many dreary propositions proved in full in the body of the text but are more interesting as well.  This is not a very interesting exercise.)
A: The basic fact to verify is that if $\cal{T}_i$, $i \in I$ is an an indexed collection of topologies on a set $X$, then their intersection $\cap_{i \in I} T = \left\{ A \subset X: \forall i \in I: A \in\cal{T}_i \right\}$ is a topology on $X$ as well. This is straightforward from the topology axioms. 
Once this is done, then the smallest topology that contains all $\cal{T}_i$ can be defined as the intersection of all topologies $\cal{T}$ that contain $\cup_{i \in I} T_i$ as a subfamily, and there is at least one (the discrete topology) so we take the intersection of a non-empty family of topologies, which is a topology as we saw above. And it is clearly the smallest one that contains all $\cal{T}_i$: if $\cal{T}$ is any such topology it is one of the topologies we take the intersection of, and thus clearly is a subset of $\cal{T}$.
Also, the largest topology contained in all $\cal{T}_i$ is simply their intersection, and this is clearly the largest topology that is contained in all of them.
