Every simple closed curve on the sphere separates it into two connected components, each having the curve as its boundary.
For a compact set homeomorphic to a circle, lifting an argument along its parametrization identifies the quotient by continuous exponentials with . By planar logarithm factorization it is also free abelian on the bounded complementary components, so exactly one exists. A proper closed arc has trivial exponential quotient, and therefore connected complement. Given a small open arc of inside any neighborhood of a point, join points from the two complementary regions by a path avoiding the remaining closed arc. The first and last intersections with occur in the small arc, proving that the neighborhood meets both regions. Thus is the common boundary of its unique bounded and unbounded regions.
A Jordan domain is the bounded complementary component of a closed simple curve in the plane. The Jordan curve theorem separates the plane into this inside component and one unbounded outside component. A conformal bijection from the disc or complex upper half-plane to this simply connected domain extends to a homeomorphism of their closed conformal boundaries by the Caratheodory boundary extension theorem.
A Jordan curve in the two-sphere is carried to an equator by a homeomorphism of the sphere; equivalently, the closures of its two complementary regions are discs. This strengthens the separation assertion of the Jordan curve theorem and controls the boundary degrees of characteristic maps for cell decompositions of a surface.
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.
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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.