Does connectedness of $cl(A)$ implies $A$ is connected?

Question

Let $A$ is subset of topological space $X$. I read proof of the fact that $A$ is connected implies $cl(A)$ is connected.I wonder if other way is true .I tried to come up with counter example but i failed .

$cl(A)$ means closure of $A$.

Answer 1

$((0,1)\cup(1,2))\subset \Bbb R$ is disconnected, but its closure $[0,2]$ is connected.

Answer 2

It's not true. $\mathbb Q$ is totally disconnected. Yet, $\overline{\mathbb Q} = \mathbb R$ is connected.

Answer 3

Consider $$A=(-1,1)\cup (1,2)$$ with the standard metric topology on real line.

$A$ is not connected while its closuer $C=[-1,2]$ is connected.