I have this problem:

Sis a set.Pis 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 onSis the power set ofS, where, by definition, every subset ofSis open.

Am I on the right track? Do I just say something along the lines ofSbeing a set of open balls and show that it's countable?