Solve Connected Set Closure: Show Not Disconnected

[metalbec]

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.

Thanks

 

[Anthony]

Call your space M. You want to show that if \mathrm{cl}(M) \subset X\cup Y, with X, Y disjoint and open then \mathrm{cl}(M) is contained in either X or Y.

Can you go from here?

 

[NateTG]

Can you solve the problem if the closure has one additional point?
How about two additional points?

 

[mathwonk]

the following version of connectedness makes all possible problems trivial:

a set is connected iff all continuous maps to the set {0,1} are constant.