One method of this proof due to Maehara derives a contradiction with the Brouwer fixed point theorem.
The Mizar and HOL Light computational systems were independently used to generate a formal proof of this theorem in 2005. One method of this proof due to Maehara derives a contradiction with the Brouwer fixed point theorem. Another proof of this theorem relies on the use of a special case of the Siefert-van Kampen theorem to demonstrate that two inclusions induce trivial homomorphisms of the fundamental groups. The theorem of its namesake and Schönflies extends this statement to stipulate that the regions involved are both homeomorphic to the circle, but Alexander's horned sphere is a counterexample to that theorem that prevents it from generalizing to dimensions greater than two. For 10 points, give this rather intuitive theorem named for a French mathematician that establishes that every non-self-intersecting closed curve in a plane divides the plane into interior and exterior regions.
The Mizar and HOL Light computational systems were independently used to generate a formal proof of this theorem in 2005.
{Jordan} curve theorem