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Published on Nov 9, 2006
A path that is homoemorphic to a circle devides a compactified plane into two pieces (inside and outside). Arthur Schönflies proved in 1906 that in this situation the inside and outside are homoemorphic.
To prove a similar statement in 3 dimensions was an open problem for many years. It was solved by James Alexander in 1928 who constructed the Alexander "Horned" Sphere, as illustrated in this video. The Alexander horned sphere is a topological space which is homeomorphic to a sphere, but inside and outside are not homeomorphic. This proves that there is no analog of Schönflies Theorem in three dimensions.
This Video was produces for a topology seminar at the Leibniz Universitaet Hannover.