Topology question

[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.