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?