Connected sets

  • Thread starter AcC
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  • #1
AcC
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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)
 

Answers and Replies

  • #3
AcC
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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?
 
  • #4
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So B is the disjoint union of nonempty open sets G and H. Can you show that [tex]A\cap G[/tex] and [tex]A\cap H[/tex] forms a disconnection of A? Thus in particacular, A is the disjoint union on nonempty open sets.
 

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