Let (M,d) be a metric space. Prove the equivalence of the following:
(i)M is disconnected
(ii)There exists two nonempty, disjoint closed sets in M whose union is M
(iii)There exists two nonempty, disjoint open sets in M whose union is M
(iv)There exists a set A in M such that emptyset != A != M, and A is both open and closed.
these are basically the definitions of a disconnected set aren't they? I have no idea how to prove these =/