Well what's your definition of disconnected? I assume that can be written as the disjoint union of open sets.

: This is clear since if (where are open) then and both of which are disjoint and closed. So then,

Suppose that is a disconnection of (with closed). Then, is closed by assumption and and since is closed it folllows that is open.

Oops...I just noticed that I supposed wrong your definition of disconnected. What is it?