If we draw a circle in the plane , we separate the plane into 2 regions , inside and outside.
On the following surfaces, determine the maximal number of circles that you can draw on the surface without creating more than one region : Sphere, Torus , Pretzel(double -doughnut), Klein bottle and a projective plane.
Honestly I don't see how is it possible to draw a circle(with radius > 0) on one of these surfaces without creating more than 1 region. A circle on a sphere is a Sphere-plane intersection. So if you draw a circle on a sphere you will create 2 regions, one above the plane and one below