Alexander Horned Sphere

The Alexander horned sphere is a pathological embedding of an ordinary two-dimensional sphere into three-dimensional Euclidean space, discovered by J. W. Alexander in 1924. Together with its interior it forms a topological ball that is simply connected, meaning every loop inside it can be shrunk to a point, but unlike the exterior of an ordinary round sphere, its exterior is not simply connected. This distinction shows that the Schoenflies theorem, which holds in two dimensions for any simple closed curve, fails to extend to three dimensions: no homeomorphism of three-dimensional space can straighten the horned sphere into a standard sphere.