Yet worse example: take $U$ to be the complement of a [pseudo-arc][1] $P$ in the unit sphere.
For any open disc $D$, such that $P \not\subset D\not\subset U$ we have $D\cap U$ is not connected. 

Take a point in $P$ as the north pole and consider the stereographic projection of $U$ to the plane.
For the obtained set $U'$ and any open topological disc $D$ such that $D\not\subset U'$ the set $D\cap U'$ is not connected. 


  [1]: http://en.wikipedia.org/wiki/Pseudo-arc