Let $E \subset X$ be a connected set. Show that $\overline{E}$ is connected.

Question

Let $E \subset X$ be a connected set. Show that $\overline{E}$ is connected.

Suppose that $\overline{E}$ is disconnected, then $\overline{E} = A \cup B$ where $A \cap B = \emptyset$ and $A,B$ are both open. Since $E \subset \overline{E}$ we must have that $E \subset A$ or $E \subset B$. Wlog assume that $E \subset A$, then $E \cap B = \emptyset$ i.e. $E \subset B^c$. This implies that $\overline{E} \subset \overline{B^c} = B^c$ but this happens if and only if $\overline{E} \cap B = \emptyset$? Is this a valid proof for this?

Answer 1

Alternative:

A topological space $(X, \tau) $ is connected iff no continuous map $f:X\to\{0, 1\}$ is onto. $[$ the target space is endowed by discrete topology $]$

Let $f:\overline{E}\to \{0, 1\}$ be a continuous function.

Goal: To show $f$ is not onto i.e $f$ is constant.

$F:=f|_E : E\to \{0, 1\}$ is also continuous and connectedness of $E$ implies $F$ is constant.

Since $E\subset \overline{E}$ is dense and the target space is Hausdorff , $F=f$ on $E$ implies $F=f$ on $\overline{E}$ implies $f$ is constant on $\overline{E}$.$($ see here and here $) $