Math 2.0

Manifolds and cell complexes Topological manifolds: basic notions Counterexample

Alexander horned sphere

Builds on Schoenflies theoremn-sphereTopological embeddingFundamental group

Counterexample

The Alexander horned sphere is a subset Σ⊂ℝ3\Sigma \subset \R^3 built by an infinite process. It is the image of a topological embedding of the 22-sphere S2S^2. Start with a ball and pull out two horns that grow towards each other. Near their tips, let each horn split into a pair of thinner horns. Let the two pairs hook through each other like two links of a chain, without touching. Repeat this splitting inside each pair, at smaller and smaller scales, infinitely often. The limit set of the surfaces is Σ\Sigma; the tips shrink to a Cantor set of points, and Σ\Sigma is still homeomorphic to S2S^2.

The closure of the bounded component of ℝ3∖Σ\R^3 \setminus \Sigma is homeomorphic to the closed ball D3D^3. The unbounded component is not simply connected. A loop around the base of one horn cannot be shrunk to a point in the complement. Each attempt to slide it off the horns is blocked by the next pair of hooked horns. So the closure of the unbounded component is not homeomorphic to the closure of the complement of a round ball, whose fundamental group is trivial.

Remarks

The example shows that the Schoenflies theorem fails in dimension 33 without its local flatness hypothesis. The sphere Σ\Sigma is not locally flat at the points of the Cantor set. J. W. Alexander described it in 1924. Gluing two copies of the closed bounded side along Σ\Sigma gives a space homeomorphic to S3S^3, a result of R. H. Bing.

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To understand this page from the ground up, read these 11 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 spaceobjectFundamental group, concept, no evidence levelFundamental groupconceptHomeomorphism, 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 theoremstatementAlexander horned sphere, counterexample, no evidence levelAlexander horned sphe…counterexample
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.
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  1. Circle concept
  2. Closed ball concept
  3. Connected space concept
  4. Euclidean space object
  5. Fundamental group concept
  6. Homeomorphism concept
  7. Topological embedding concept
  8. n-sphere object
  9. Jordan curve theorem statement
  10. Schoenflies theorem statement
  11. Alexander horned sphere counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Alexander_horned_sphere
About this page
TagQQEASX79RW
URIm2://QQEASX79RW
Kindcounterexample
WritingW0 stub
MSC57M30, 57N45
UsesSchoenflies theorem, n-sphere, Topological embedding, Fundamental group
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Changed2 November 2026

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