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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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    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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    @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