Janiszewski's theorem

Short description: 2 points connected in the plane while avoiding 1 of 2 intersecting subsets can avoid both

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 and in complex function theory.

References

• Bing, R. H. (1983), The Geometric Topology of 3-Manifolds, Colloquium Publications, 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, 15, Vandenhoeck & Ruprecht 
• Pommerenke, C. (1992), Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften, 299, Springer, ISBN 3540547517