Alexander horned sphere

A wild embedding of a two-dimensional sphere in three-dimensional space, constructed with interlocking horns, whose complement is not simply connected.

Alexander's horned ball: The wild three-dimensional region bounded by Alexander's horned sphere. It is the enclosed region in the classical construction, with the sphere as its boundary.
Topological embedding: A continuous map that identifies a space with a subspace while preserving its topology. The horned sphere is a sphere embedded in three-dimensional space.
James Waddell Alexander II: An American mathematician whose work included major contributions to topology and knot theory. He introduced the horned-sphere construction in 1924.
Jordan curve theorem: The theorem that every simple closed curve in the plane separates the plane into an inside and an outside. The horned sphere extends the separation theme to a setting where complement topology becomes subtler.
Fundamental group: An algebraic group that records loops in a space up to continuous deformation. The complement’s nontrivial fundamental group formalizes the obstruction created by the horns.
Two-sphere: The surface of a ball, consisting of points at a fixed distance from its center. The construction begins with this ordinary surface before deforming it wildly.
Combinatorial topology: The study of topological spaces through discrete structures such as complexes and triangulations. The example belongs to the period when topology probed the limits of geometric intuition.
Alexander duality: A theorem relating the homology of a subspace’s complement to the cohomology of the subspace. It provides an algebraic framework for studying complements of embedded spheres.
Simply connected space: A path-connected space in which every loop can be continuously contracted to a point. The sphere’s complement fails precisely this property.
Complement (topology): The part of an ambient space left after a specified subspace is removed. The defining feature concerns the topology of the space outside the embedded sphere.

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