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?