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. 

Yes, it would have to contain a point both inside and outside the circle.
Let $P$ be a point of the circle and $U_\epsilon$ an epsilon neighborhood of $P$ centered at $P$ and contained in the region.
Then the ray from the origin containing $P$ contains points of the region at distances $1-\frac{\epsilon}{2}$ and $1+\frac{\epsilon}{2}$ from the origin.
