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Suppose that C, D are connected subsets of a topological space T such that $\bar{C} \cap \bar{D} \neq \emptyset$. Is it true that $C \cup D$ is necessarily connected?

I think I have a counter example for this:

Take the intervals $C = (0,1)$ and $D = (1,2)$ their closure is $[0,1]$ and $[1,2]$ which gives $\bar{C} \cap \bar{D} = \{1\} \neq \emptyset$ but clearly $(0,1) \cup (1,2)$ is disconnected. Am I right?

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    @user26069: You might consider taking the second part of your question (beginning with "I think...") and making it an answer on your own question. This is allowed, and in fact explicitly encouraged, on the StackExchange network.2012-03-04

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To get this off the Unanswered list, I’m posting the OP’s correct counterexample as CW:

Take the intervals $C = (0,1)$ and $D = (1,2)$ their closure is $[0,1]$ and $[1,2]$ which gives $\bar{C} \cap \bar{D} = \{1\} \neq \emptyset$ but clearly $(0,1) \cup (1,2)$ is disconnected.