Connected sets

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 cant find a solution exactly.. ım not sure that we work in topological space (and are G and H open in X)

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.