I'm trying to teach myself topology from Munkres' book (first edition, it's over 30 years old!) and it's going very slowly. I'm having some difficulty proving the following:
Let be a collection of topologies on . Show that there is a unique smallest topology containing all the collections , and a unique largest topology contained in all .
This is part b in a three part problem. The first part was to prove that if is a collection of topologies on that is a topology on , which I was able to do. So, imagine that result will be useful in what I'm trying to prove here.
Oh wow, I can't believe I missed that first part. I had pretty much gotten to the end, but some how I missed the fact that is finer than any other topology contained in every . It seems so obvious now!
Okay, for the second part, let's see if I get this.
Since the discrete topology contains every subset of , it clearly contains . So, the intersection of all topologies containing , call it , is non-empty. If is some topology containing , then by definition, . Thus, is the smallest topology containing all topologies in .
Does that look right?
Yes. The reason why the intersection is non-empty is because is an example of a topology containing .
Originally Posted by spoon737