For example, take a sphere. Draw a small circle around the point that is the North pole. Now you can expand the circle by pulling it down and stretching it until it fits around the equator. If you pull it down further, it becomes a small circle encircling the South pole. Does this circle still encircle the North pole?

Basically, it bothers me that if you have a point on the surface of a sphere, and a circle next to it (but not enclosing it), you can stretch the circle the other way around the sphere until it encloses the point! If you have a basketball and draw a point on it, and get a rubber band and place it next to the point (thereby enclosing an area next to the point), then you can stretch one half of the rubber band around the equator to enclose the point.

There are some things like indices that one needs to calculate by drawing a loop and counting the singularities inside, but I don't really know if the loop encloses the singularity or not, because stretching the loop without crossing any other singularities allows you to ecnlose any point you want.