Alexander horned sphere
Pathological embedding of the sphere in 3D space
Alexander horned sphere
The Alexander horned sphere is a pathological embedding of the 2-sphere into 3-dimensional Euclidean space. The topological object was discovered by J. W. Alexander (1924). It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. However, the exterior is not simply connected, unlike the exterior of the usual round sphere. The Schoenflies theorem in 2D states that any simple closed curve in the plane can be extended to a homeomorphism of the entire plane. If this held in 3D, the exterior of any embedded sphere would have to be homeomorphic to the exterior of a standard sphere. The Alexander horned sphere proves this is false because the exterior of a standard sphere is simply connected, while D e x...
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