Proof of disconnected sets
I have to show that given any topological space Y, if a closed subset of Y is disconnected then it must be the union of two disjoint closed sets.
i.e. C= U union V where U and V are closed nonempty sets, and their intersection is empty.
My proof so far:
Let C be a closed subset of Y. Assume C is disconnected. Then there exists two nonempty subsets U,V such that U does not equal C and V does not equal C. Im not sure if I can assume that U= C(V) (complement of V) and V= C(U). If so that would imply that they are disjoint. So I need to show that if U is relatively closed that the complement is relatively open. This implies that if V is relatively open and closed than U is also relatively open and closed. But this is where I get confused because I have to show that C=U union V where U and V are closed disjoint sets. I'm stuck.