Every simple closed curve on the sphere separates it into two connected components, each having the curve as its boundary.
The outer boundary of a compact planar set is the boundary of the unbounded component of its complement. Its filled hull is . When the outer boundary is a simple closed curve, the hull is its closed interior. For a simply connected open domain , containment of a compact trace and of its filled hull are equivalent. This lets outer-boundary pushforwards retain simply connected domain-containment information.
Articles by others on the same topic
The Jordan Curve Theorem is a fundamental result in topology, a branch of mathematics that studies properties of spaces that are preserved under continuous deformations. The theorem states that any simple closed curve in a plane (a curve that does not intersect itself and forms a complete loop) divides the plane into two distinct regions: an "inside" and an "outside.