>>11931733>>119317602. there are at least two components
this uses for example the winding number. for a point , the integer measures how many times does wind around . it's a generalization of that thing from complex analysis. you need two things:
A. is locally constant in . It follows that points with distinct winding numbers lie in distinct components of .
B. you can compute it as follows. draw a ray emanating from which is transverse to . is the signed intersection number of with .
by A. we just need to find two points with distinct winding numbers. is compact, so if you pick very far from it, because you can find an arc which doesn't intersect at all. now we need a point with . since is compact, there is a point with the -coordinate minimal (let's say). pick to be slightly to the right from that point and pick to be the negative -direction. now crosses exactly once, so .
