The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove.
22.
Replacing ? by a homotopic curve, it may be assumed that ? is a smooth Jordan curve ? with non-vanishing derivative.
23.
Some examples are the Hahn Banach theorem, K�nig's lemma, Brouwer fixed point theorem, G�del's completeness theorem and Jordan curve theorem.
24.
The proof of the Jordan curve theorem for " differentiable curves " is not difficult, and can be done using mathematics available to Gauss.
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One proof of the impossibility of finding a planar embedding of " K " 3, 3 uses a case analysis involving the Jordan curve theorem.
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In fact, it's a simple corollary of the Jordan curve theorem ( which basically says every simple closed curve has an inside and outside ).
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:: : Smale is talking about the Jordan curve theorem, which states that a closed continuous curve in the plane has an inside and an outside.
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Completing the curve to a Jordan curve by adding part of the boundary of the smaller disk, the formula reduces to the planar Green-Stokes formula.
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The first formal proof of the Jordan curve theorem was created by in the HOL Light system, in January 2005, and contained about 60, 000 lines.
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He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof, many now also consider Jordan's original proof rigorous.
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