I'm struggling with something that I suspect is very basic. How do I should that the closure of a connected set is connected? I think I need to somehow show that it is not disconnected, but that's where I'm stuck.
Call your space [tex]M[/tex]. You want to show that if [tex]\mathrm{cl}(M) \subset X\cup Y[/tex], with [tex]X, Y[/tex] disjoint and open then [tex]\mathrm{cl}(M)[/tex] is contained in either [tex]X[/tex] or [tex]Y[/tex].