LetAbe connected subset ofXand letA⊂B⊂ cl(A).Show thatB is connectedand hence, in particuar, cl(A) is connected.

Hint: (Use) Let G∪H be a disconnection ofAand letBbe a connected subset ofAthen we see that eitherB∩H=∅or B∩G=∅, and so eitherB⊂GorB⊂H.