jordan-polygon-theorem (1) - Computational Topology (Jeff...

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Unformatted text preview: Computational Topology (Jeff Erickson) The Jordan Polygon Theorem The fence around a cemetery is foolish, for those inside cant get out and those outside dont want to get in. Arthur Brisbane, The Book of Today (1923) Outside of a dog, a book is mans best friend. Inside of a dog, its too dark to read. attributed to Groucho Marx 1 The Jordan Polygon Theorem The Jordan Curve Theorem and its generalizations are the formal foundations of many results, if not every result, in surface topology. The theorem states that any simple closed curve partitions the plane into two connected subsets, exactly one of which is bounded. Although this statement is intuitively clear, perhaps even obvious, the generality of the phrase simple closed curve makes the theorem incredibly challenging to prove formally. According to most classical sources, even Jordans original proof of the Jordan Curve Theorem [ 3 ] was flawed; most sources attribute the first correct proof to Veblen almost 20 years after Jordan [ 6 ] . (But see also the recent defense and updated presentation of Jordans proof by Hales [ 2 ] .) However, we can at least sketch the proof of an important special case: simple polygons. Polygons are by far the most common type of closed curve in practice, so this special case has immediate practical consequences. Moreover, most proofs of the Jordan Curve Theorem rely on this special case as a key lemma. (In fact, Jordan dismissed this special case as obvious.) 1.1 First, A Few Definitions A homeomorphism is a continuous function h : X Y with a continuous inverse h- 1 : X Y . Two topological spaces are homeomorphic (or topologically equivalent ) if there is a homeomorphism from one to the other. A simple path is a subset of the plane that is homeomorphic to the unit interval [ 0,1 ] R , or equivalently, the image of a continuous injective function from [ 0,1 ] into the plane. 1 A subset X of the plane is (path-)connected if for any two points in X , there is a simple path in X from one point to the other. A connected component of X is a maximal path-connected subset of X . Similarly, a simple closed curve is a subset of the plane that is homeomorphic to the unit circle S 1 : = { ( x , y ) R 2 | x 2 + y 2 = 1 } , or equivalently, the image of a continuous injective function from S 1 into the plane. The full-fledged Jordan curve theorem states that for any simple closed curve C in the plane, the complement R 2 \ C has exactly two connected components. Finally, a simple path or closed curve is polygonal if it is the union of a finite number of line segments (called edges ). An endpoint of an edge is called a vertex . A polygonal path A simple polygonal closed curve is also called a simple polygon ....
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jordan-polygon-theorem (1) - Computational Topology (Jeff...

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