Little confusion about connectedness Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$.
So, $\overline X$ is also connected , as closure of connected set is connected.
Now if we notice about the set $\overline X$ then 
$$\overline X=X\cup B$$
where , $B=\{(0,y):-1\le y\le 1\}$
Now , $X$ and $B$ both are connected , and $X\cap B=\emptyset$ . So, $\overline X$ is disconnected.
Where my mistake ??

I think that in your reasoning there is two incorrectness:


*

*We have $\overline{X}=X\cup B\cup\{(1,\sin(1))\}$.

*The implication : 
$X$ and $B$ are connected and $X\cap B=\emptyset$ $\Rightarrow$ $X\cup B$
is not connected 
is false. For a counterexample one can see that $\{0\}$ and $(0,1]$ are connected and disjoint, and $\{0\}\cup(0,1]=[0,1]$ is also connected.
