Let $A$ be a connected subspace of $X$. If $A\subset B\subset\bar{A}$, then $B$ is also connected.

Question

Let $A$ be a connected subspace of $X$. If $A\subset B\subset\bar{A}$, then $B$ is also connected.

My attempt: Let $A$ be a connected subspace of $X$ and let $A\subset B\subset\bar{A}.$ Suppose that $B=C\cup D$ is a separation of $B$. So $A\subseteq C$ or $A\subseteq D$. Suppose $A\subseteq C$. Then $\bar{A}\subseteq \bar{C}$.

So how can I continue the attempt, may you help?

Answer 1

You're almost there. If $\overline A \subset \overline C$, we know $\overline C = \overline A$, because the other inclusion follows from $C\subset B\subset\overline A$ (Take the closure).

It follows then that $D=B\setminus C = \emptyset$, which is a contradiction.

Answer 2

Assume that $B$ is not connected.

Since $A$ is connected,then $\overline{A}$ is connected.

Assume that $C,D \neq \emptyset$ is a separation of $B$ where $D \cap \bar{C}=\bar{D} \cap C=\emptyset$

Then since $A$ is connected,then $A \subseteq C$ (as in your post)

Now, every point of $B$ is a limit point of $A$ and hence a limit point of $C$ (since $A \subseteq C$). Therefore no point of $B$ belongs to $D$ (since $D$ does not contain limit points of $C$) and this is wrong since $D \neq \emptyset$