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 ??

The set $B$  is NOT open, so this is not a partition of $\overline X$ in open sets.
