TITLE: basic doubt about connectness - general topology
QUESTION [1 upvotes]: Let $E \subset R^n$, where $E^c$ is disconnected. 
Then exists $U,V \subset R^n$,     $U,V \neq \emptyset$    disjoint,  and open relative to $E^c$ , with $E^c = U \cup V.$
We have too $U \cap \overline{V} = \emptyset = V \cap \overline{U} $ . My book says :
Exists a ball $B \subset R^n$ such that $B$ intersects both ${\overline{U}}^c$ and ${\overline{V}}^c$. 
Drawing a picture is easy to see this last affirmation, but i dont know how to prove that. Someone can give a hint ? Any help is apreciated.
Thanks in advance!

REPLY [0 votes]: $\overline U^c$ and $\overline V^c$ are open so if this ball don't exist, 
we have: (1) $\overline U^c \cap \overline V^c = \emptyset$ hence (deMorgan's law) $\overline U \cup \overline V = \mathbb R = \overline{U \cup V}$. So $U \cup V$ is dense in $\mathbb R$. But we have $\overline V \subset \overline U^c$ and $\overline U \subset \overline V^c$.
(from condtitons between $U$, $\overline U$ , $V$ and $\overline V$)
$\overline U^c$ and $ \overline V^c$ are open and (1) - contradiction.
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