# Under Poincaré-Bendixson hypothesis, can a $\omega(p)$-limit set be like this picture?

Last class we were proving Poincaré-Bendixson theorem in $$\mathbb R^2$$ which states that:

Assume that the positive orbit $$\mathcal O^+(p)$$ is contained in a compact subset $$K$$ of the planar domain $$D$$ of the differential equation $$x'=X(x)$$. Assume further that $$X$$ has only finitely many fixed points in $$K$$. Then one of the following is satisfied:

a) $$\omega(p)$$ is a periodic orbit;

b) $$\omega(p)$$ is a single fixed point

c) $$\omega(p)$$ consists of a finite number of fixed points, together with a finite set of orbits such that for each orbit its $$\alpha$$-limit set is a single fixed point and its $$\omega$$-limit set is also a single fixed point.

During the proof, he stopped the class and drew the following picture in the blackboard: and asked, can $$\omega(p)$$ be like that?

So I assume that he asked that in the context given by the hypothesis of Poincaré-Bendixson theorem. Also he intended to picture with those dots the singularities of the field. I want to say that the answer is no, because I think that every singularity must be connected by an orbit, which does not happen in this picture. But the problem is: I can't justify that. I've read the poincaré-bendixson demonstration quite a few times, but I can't find this justificative there. Any insight would be very helpful. Thank you