Alexander horned sphere

The exterior is not simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere.

This shows that the Jordan–Schönflies theorem does not hold in three dimensions, as Alexander had originally thought.

Alexander also proved that the theorem does hold in three dimensions for piecewise linear/smooth embeddings.

The closure of the non-simply connected domain is called the solid Alexander horned sphere.

The solid Alexander horned sphere is an example of a crumpled cube; i.e., a closed complementary domain of the embedding of a 2-sphere into the 3-sphere.