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for this question there is something in my mind but I could not bring them one pieces which gives the solution. Probably I am overlooking something but I do not know what it is. can you please share your idea or answer...

(X,T) is a topological space. A and B are connected subspace of X. show that if the intersection of closure A and B are non-empty. Then union of A and B are connected as well.

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    Quick clarification: do you mean $\bar{A}\cap B\neq\emptyset$ or $\bar{A}\cap\bar{B}\neq\emptyset$?2012-11-30
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    (I mention this because the claim is false under the assumption that $\bar{A}\cap\bar{B}\neq\emptyset$ - take $A=(0,1)$ and $B=(1,2)$.)2012-11-30
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    I mean the first one just the closure of A.2012-11-30

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