Topological theorem asserting that every simple closed curve (a continuous injective image of the circle) in the plane separates the plane into exactly two regions: an interior bounded region and an exterior unbounded region, with the curve as their common boundary.

Principle

A simple closed curve in the plane is homeomorphic to the circle and its complement in R^2 has exactly two connected components; one component is bounded (the inside) and the other unbounded (the outside), and the curve equals the boundary of each component.

Demonstration

For an ordinary circle the theorem is immediate: the disk is the interior and the complement is the exterior. More strikingly, even highly wiggly continuous simple closed curves (including nowhere differentiable examples) still separate the plane in the same way, by the Jordan Curve Theorem.

Misapplication

Assuming the separation result holds for curves with self-intersections, for non-planar embeddings, or in higher dimensions without modification; treating the theorem as giving a constructive way to find the inside in pathological cases is also a misuse.

Consequence

Provides a rigorous foundation for intuitive notions of inside and outside in plane topology, underpins results in planar graph theory, complex analysis (contour integration), and supports arguments using winding number and index.

Reversal

A non-simple curve (one with self-intersections) need not separate the plane into exactly two regions; similarly, in three dimensions a closed curve does not generally separate space into two regions (Alexander duality differs in higher dimensions).

Boundary

Applies to continuous embeddings of the circle into the plane (simple closed curves); excludes curves with self-intersections, embeddings into other surfaces without checking genus, and higher-dimensional analogues that require different hypotheses.

Semantic Tension

Tension exists between the intuitive planar notion of 'inside' and the topological formulation: for pathological simple curves the interior may be topologically complicated, so geometric intuition about convex or smooth boundaries can be misleading.

Synthesis

The Jordan Curve Theorem formalizes the intuitive fact that a simple closed planar curve divides the plane into two complementary regions with the curve as common boundary; it is a fundamental separation result whose validity for highly irregular curves is a key insight of topology.