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Jordan Curve Theorem

If J is a simple closed curve in R^2, then the Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966) states that R^2-J has two components (an "inside" and "outside"), with J the boundary of each.

The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher (2002, p. 169), or in classic texts such as Spanier (1966). Recently, a proof checker was used by a Japanese-Polish team to create a "computer-checked" proof of the theorem (Grabowski 2005).

See also

Jordan Curve, Schönflies Theorem

Portions of this entry contributed by Dmitrii Pasechnik

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References

Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, p. 68, 1995.Grabowski, A. "Culmination of a Complete Proof of the Jordan Curve Theorem." 2005. http://markun.cs.shinshu-u.ac.jp/mizar/jordan/jordancurve-e.html.Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002. http://www.math.cornell.edu/~hatcher/AT/ATpage.html.Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 14, 1996.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 9, 1976.Spanier, E. H. Algebraic Topology. New York: McGraw-Hill, 1966.

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Jordan Curve Theorem

Cite this as:

Pasechnik, Dmitrii and Weisstein, Eric W. "Jordan Curve Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JordanCurveTheorem.html

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