Theorem

Schoenflies Theorem

Topology

Every simple closed curve in the plane can be extended to a homeomorphism of the whole plane carrying it to a standard circle, so that the interior and exterior it bounds are each homeomorphic to a disk and its complement. Named for Arthur Schoenflies, it strengthens the Jordan curve theorem in two dimensions.

Facts

Statement

Every simple closed curve in the plane separates the plane into a bounded interior and an unbounded exterior, and there is a homeomorphism of the plane carrying the curve onto a standard circle, taking interior to interior and exterior to exterior. 1

Proof Year

1906 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Topology

Sources

1. Schoenflies Theorem (Wikipedia)

Wikimedia Foundation

lead paragraph, original formulation sentence
The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the 'inside') bounded and the other (the 'outside') unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
references section, Schoenflies 1906 citation
Schoenflies, A. (1906), Beitrage zur Theorie der Punktmengen III, Mathematische Annalen, 62 (2): 286-328

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Schoenflies Theorem

Has Statement Form

In Branch