Verifying a fact about Connectedness

Question

Munkres gives a proof of the following claim, I'm just checking if mine is okay:

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

Pf: $A$ is connected so it is closed. Thus $A \subset B \subset A \implies A = B$. So $B$ is connected.

Answer 1

"$A$ is connected so it is closed" is totally wrong !

Example : $]0,1[$ is connected in $\mathbb{R}$, but is not closed.

Answer 2

Let $f: B \to \{0,1\}$ be continuous, where the doubleton has the discrete topology. $f|A$ is constant by connectedness of $A$ and the assumption $A \subset B \subset \overline{A}$ implies that $A$ is dense in $B$ and as the image is Hausdorff, $f$ must be constant too as it coincides with a constant map on a dense subset and hence on the whole space. Ergo, $B$ is connected.