Horned sphere

Let A,B ⊆ IRⁿ be closed and homeomorphic as topological spaces equipped with the topology induced by the Euclidean one of IRⁿ. It is known that any homeomorphism h:A—>B extends to a homomorphism φ:IR²ⁿ—>IR²ⁿ. It follows that IR²ⁿ \ A is homeomorphic to IR²ⁿ \ φ(A) = IR²ⁿ \ h(A) = IR²ⁿ \ B . In general, however, it is not true that for any m we have IRᵐ \ A ≅ IRᵐ \ B. A notable (but quite pathological) example is given by the Horned sphere Σ ⊆ IR³. It is homeomorphic to the sphere S² but IR³ \ Σ is not homeomorphic to IR³ \ S². Other notable examples can be found in Rushing’s “Topological Embedding”. A representation of the Horned sphere is given in the figure below.