What does it mean for a curve to "enclose" points on a sphere?

Question

I'm studying index theory from Strogatz "Nonlinear Dynamics" book. In it, he asks a question about whether a closed curve contains fixed points with indices summing to +1 on non-planar surfaces. For the torus and cylinder, it's easy to show that the flow can contain only closed orbits, and thus no fixed points. However, for a sphere, there must be a fixed point (by hairy ball theorem). My problem is that I can't tell what it means for a path to "enclose" a point on a sphere. For a plane, it's easy, since a closed orbit will separate the plane into a bounded region and an unbounded one, to the bounded one is logically "enclosed". However, for a sphere, the two regions are both bounded, and could potentially be the same size. Does that mean that a closed path on a sphere "encloses" all points on the sphere that aren't on the path?