# 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

## Answers and Replies

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?

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

Science Advisor
Homework Helper
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.