I seem to recall that you can say a set is closed if there exists a sequence that converges to a limit point of that set...obviously that is not correct but the idea is that you can deduce a set is closed because of the existence of some converging sequence, something along those lines. I think it was and "if and only if" theorem, so that the set being closed also gives you information about the sequences.
Does anyone know the proper theorem for this concept?