Second countable, countable, proof...
I have this problem:
S is a set. P is the discrete topology on S. Prove that (S,P) is second countable iff S is a countable set.
Can somebody expand on this? (I assume it's basically what I'm trying to prove):
"Every discrete space is first-countable; it is moreover second-countable if and only if it is countable."
Is this way of thinking correct (I'm an undergrad):
A topology is second countable if it has a second countable base.
A base is second countable if it contains a countable number of open sets.
The discrete topology on S is the power set of S, where, by definition, every subset of S is open.
Am I on the right track? Do I just say something along the lines of S being a set of open balls and show that it's countable?