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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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HINT: Suppose that $U$ and $V$ are a separation of $A\cup B$: $U$ and $V$ are non-empty clopen subsets of $A\cup B$, $U\cup V=A\cup B$, and $U\cap V=\varnothing$. Without loss of generality assume that $A\cap U\ne\varnothing$. Use the connectedness of $A$ and $B$ to show that $A=U$ and $B=V$, and arrive at the contradiction that $$(\operatorname{cl}A)\cap B=(\operatorname{cl}U)\cap V=\varnothing\;.$$

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    @icurays1: Yes, I had too many closures in there; it’s fixed now, assuming that I’m reading the OP’s *intersection of closure A and B* correctly. Thanks.2012-11-30
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    Accidentally deleted the comment that led to that, whoops.2012-11-30
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    @BrianM.Scott: there is one point I dont understand can you explain it more clearly? why we are assuming intersection of ((A) and (U)) are non-empty?and how does this expression contribute to solution..Thnx2012-11-30
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    @Ridvan: Since $U\cup V=A\cup B$, at least one of $U$ and $V$ must intersect $A$, and I might as well call that one $U$.2012-11-30