Connected_Sets

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Definition

A set $E \subseteq \mathbb{R}$ is disconnected if it can be written as $E = A \cup B$ where $A$ and $B$ are nonempty, disjoint sets such that $\overline{A} \cap B = \emptyset$ and $A \cap \overline{B} = \emptyset$.

A set $E$ is connected if it is not disconnected.

Theorem: A set $E \subseteq \mathbb{R}$ is connected if and only if it is an interval.

Proof

($\Leftarrow$) Suppose $E$ is an interval and $E = A \cup B$ is a separation. Let $a \in A$ and $b \in B$ with $a < b$. Since $E$ is an interval, $[a,b] \subseteq E$.

Let $c = \sup(A \cap [a,b])$. Then $c \in \overline{A}$, so $c \notin B$, meaning $c \in A$. But then $(c, c+\epsilon) \cap B \neq \emptyset$ for small $\epsilon$, so $c \in \overline{B}$, contradicting $A \cap \overline{B} = \emptyset$.

($\Rightarrow$) If $E$ is not an interval, there exist $a < c < b$ with $a, b \in E$ but $c \notin E$. Then $A = E \cap (-\infty, c)$ and $B = E \cap (c, \infty)$ form a separation.