Jordan curve theorem

Jordan curve theorem

1. (topology) The theorem that states that a simple closed curve (Jordan curve) divides the plane into precisely two distinct areas.
   • 1995, William Fulton, Algebraic Topology: A First Course, Springer, page 343:

     There is a vast generalization of the Jordan curve theorem to higher dimensions.

   • 2001, Theodore Gamelin, Complex Analysis, Springer, page 249:

     The Jordan curve theorem asserts that a simple closed curve in the complex plane divides the plane into exactly two connected components, one bounded (the "inside") and the other unbounded (the "outside" ).

   • 2004, Nelson G. Markley, Principles of Differential Equations, Wiley, page 193:

     The principal reason that orbit behavior in the plane is more limited than in higher dimensions is the Jordan curve theorem. Local sections will be the primary tool used to exploit the Jordan curve theorem.

2. Used other than figuratively or idiomatically: see Jordan curve,‎ theorem.
   • 2009, Josef Šlapal, Jordan Curve Theorems with Respect to Certain Pretopologies on ${\displaystyle \mathbb {Z} ^{2}}$, Srecko Brlek, Christophe Reutenauer, Xavier Provençal (editors), Discrete Geometry for Computer Imagery: 15th IAPR International Conference, Springer, LNCS 5810, page 252,

     Some known Jordan curves in the basic topology are used to prove Jordan curve theorems that identify Jordan curves among simple closed ones in each of the four quotient pretopologies.