I saw a proof for Jordan curve theorem in this post Triangulation of a simple polygon (elementary proof?). However, every path is the limit of polygons. Doesn't it mean that you can proove Jordan curve theorem using this polygon approximation, if you take a sequence of simple polygons converging to the path? I tried to prove it myself but it was very technical and I didn't managed to do that, so I thought maybe this approach is wrong.
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Take a look at these examples to see why it's not easy, which also means that it's great that you didn't get an easy wrong proof! You could have asked in a comment on my answer. But you're lucky that I saw your question by chance. =) – user21820 Feb 19 '22 at 17:32
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Yes you can prove the Jordan curve theorem for continuous closed simple curves by approximation by polygons. This is how the original proof of Jordan proceeded.
But there is a difficulty. If you take finitely many points on a simple closed curve, the resulting polygon can have self-intersections, even though the curve has not. This comes from the fact that a continuous curve can oscillate a lot locally. So you need to remove the loops that appears in your approximation and then it works. This is quite technical and this is the reason that other proofs were devised.
coudy
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2Another issue is that a uniform limit of injective maps need not be injective. – Moishe Kohan Feb 18 '22 at 15:50