Proving Connectivity of B & cl(A)

[AcC]

Homework Statement

Let A be connected subset of X and let A ⊂ B ⊂ cl(A). Show that B is connected and hence, in particuar, cl(A) is connected.

Hint: (Use) Let G∪H be a disconnection of A and let B be a connected subset of A then we see that either B∩H=∅ or B∩G=∅, and so either B⊂G or B⊂H.




2. The attempt at a solution
ı try to use contrapositive method but ı can't find a solution exactly.. ım not sure that we work in topological space (and are G and H open in X)

 

[grey_earl]

Does this help you?
http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=1385.0001

 

[AcC]

Thanks grey_earl but I want to show if A is connected set and A ⊂ B ⊂ cl(A) then B is connected..
I have to show that ""Suppose B=G∪H is a disconnection of B, then using the fact that B ⊂ cl(A) prove that contradict the given A is connected""
How can I show this?

 

[micromass]

So B is the disjoint union of nonempty open sets G and H. Can you show that $$A\cap G$$ and $$A\cap H$$ forms a disconnection of A? Thus in particacular, A is the disjoint union on nonempty open sets.