Let be an infinite set and let

How big is ?

What if we restrict to be the set of all the topologies on up to homeomorphism?

It's pretty clear that an upper bound (in either case) is . In the first case, is definitely a lower bound, because for each , we can define . In the second case, it's got to be infinite, because you can take a countable subset and define the topologies and .

Beyond this, I'm not too sure.