Math 2.0

Manifolds and cell complexes Theorem

Schoenflies theorem

Builds on Jordan curve theoremTopological embeddingHomeomorphismClosed balln-sphere

Statement

  1. Let C⊂ℝ2C \subset \R^2 be a Jordan curve. Then there is a homeomorphism h ⁣:ℝ2→ℝ2h \colon \R^2 \to \R^2 with h(C)=S1h(C) = S^1, the unit circle. In particular, the closure of the bounded component of ℝ2∖C\R^2 \setminus C is homeomorphic to the closed disk D2D^2.
  2. (Generalized Schoenflies theorem.) Let n≥2n \ge 2 and let e ⁣:Sn−1×[−1,1]→Sne \colon S^{n-1} \times [-1, 1] \to S^n be a topological embedding, where SnS^n is the nn-sphere. Let Σ=e(Sn−1×{0})\Sigma = e(S^{n-1} \times \set{0}). Then Sn∖ΣS^n \setminus \Sigma has two components, and the closure of each is homeomorphic to the closed ball DnD^n.

Remarks

The hypothesis in part 2 says that the sphere Σ\Sigma is bicollared. For a codimension-one sphere this is equivalent to being locally flat. It cannot be dropped for n=3n = 3: the Alexander horned sphere is an embedded 22-sphere one of whose complementary regions is not simply connected, so its closure is not a ball. In the plane no such hypothesis is needed, because every Jordan curve is tame. One proof of part 1 uses the Riemann mapping theorem and Carathéodory's extension theorem for the interior; part 2 was proved by Morton Brown in 1960. For smooth embedded spheres in ℝ4\R^4, whether the closed inside is always a smooth ball is open.

Going deeper

For a student

In higher dimensions, the naive analogue fails: there are wild embeddings of S2S^2 in ℝ3\R^3 whose outside is not the outside of a ball. The generalized theorem shows that all such wildness lives at points where the sphere is not locally flat. It is a basic tool in topological manifold theory, for example for the annulus theorem and for showing that the connected sum is well defined.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Circle, concept, no evidence levelCircleconceptClosed ball, concept, no evidence levelClosed ballconceptConnected space, concept, no evidence levelConnected spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHomeomorphism, concept, no evidence levelHomeomorphismconceptTopological embedding, concept, no evidence levelTopological embeddingconceptn-sphere, object, no evidence leveln-sphereobjectJordan curve theorem, statement, no evidence levelJordan curve theoremstatementSchoenflies theorem, statement, no evidence levelSchoenflies theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circle concept
  2. Closed ball concept
  3. Connected space concept
  4. Euclidean space object
  5. Homeomorphism concept
  6. Topological embedding concept
  7. n-sphere object
  8. Jordan curve theorem statement
  9. Schoenflies theorem statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Schoenflies_problem
About this page
Tag3VX06XRVZG
URIm2://3VX06XRVZG
Kindstatement, theorem
WritingW0 stub
MSC57N50, 57N45
StatusThis copy of the site has no build data. Run m2 status 3VX06XRVZG.
AboutJordan curve theorem
UsesJordan curve theorem, Topological embedding, Homeomorphism, Closed ball, n-sphere
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Changed2 November 2026

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