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.
That is what you are really asked to prove.
As a lemma (previous theorem) you should have proved that if is a connected set and where are separated sets then or .
Two set are said to be separated if neither is empty and neither contains a point nor a limit of the other.
Now it is quite easy to prove the lemma. If the conclusion is false then is a separation of .
But is connected. So that is a contradiction.