A region in the plane that has to intersect unit circle If a region (meaning open, connected and non-empty subset) of the plane intersects the unit circle, does it mean it has to contain points both inside and outside of the unit circle?
I wanted to make sure that there is no weird construction that would provide a counter-example to containing points from both inside and outside?
Thank you. 

Here is a sketch of a proof. I imagine you will be able to fill in the gaps.
Suppose that $U$ is a region you describe which intersects a point $p$ on the unit circle. Suppose without loss of generality that $p = 1 $. Since $U$ is open there is some disk  contained in $U$ centred at $1$. This disk must contain points in the interval $(-1,1)$.
