Fix any number $\delta>0$ and put $A = \{x \in \mathbb{R}: \left|x-3\right|<\delta\}$ and $B = \{x \in \mathbb{R}: \left|x-3\right|>\delta \}$. Prove that $C=A \cup B$ is not a connected set.
Definition of connected: Let $E$ be a subset of a metric space $(S,d)$. The set $E$ is disconnected if there are disjoint open subsets $U_1$ and $U_2$ in $S$ such that $E\subseteq U_1 \cup U_2$, $E\cap U_1\neq\emptyset$ and $E\cap U_2 \neq \emptyset$.
A set $E$ is connected if it is not disconnected.
Intuitively, this obviously makes sense. I just don't get how to prove this is proof form.
Best Answer
A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. But $A$ and $B$ are both open.