TITLE: Must the intersection of connected sets be connected?
QUESTION [11 upvotes]: Must the intersection of two connected sets be connected?
I believe the answer is no, but I am not entirely sure. I think a counter example would be a set that intersects another set in more than one area, yet those intersections are disjoint. (Think of a cylinder that extends through one end of a donut, the hole in the middle, and then the other end of the donut).

REPLY [27 votes]: No. Consider the intersection of the line segment and the circle in $\varnothing$.