Janiszewski's theorem

In mathematics, Janiszewski's theorem, named after the Polish mathematician Zygmunt Janiszewski, is a result concerning the topology of the plane or extended plane. It states that if A and B are closed subsets of the extended plane with connected intersection, then any two points that can be connected by paths avoiding either A or B can be connected by a path avoiding both of them. The theorem has been used as a tool for proving the Jordan curve theorem^[1] and in complex function theory.

References

^ "Proof of the Jordan curve theorem" (PDF).

Bing, R. H. (1983), The Geometric Topology of 3-Manifolds, Colloquium Publications, vol. 40, American Mathematical Society, ISBN 0-8218-1040-5
Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
Pommerenke, C. (1992), Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften, vol. 299, Springer, ISBN 3540547517

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[1] "Proof of the Jordan curve theorem" (PDF).

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