In topology, the Phragmén–Brouwer theorem, introduced by

Lars Edvard Phragmén Lars Edvard Phragmén (2 September 1863 Örebro – 13 March 1937) was a Swedish mathematician. The son of a college professor, he studied at Uppsala then Stockholm, graduating from Uppsala in 1889. He became professor at Stockholm in 1892, afte ...

and

Luitzen Egbertus Jan Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...

, states that if ''X'' is a

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locally connected topological space, then the following two properties are equivalent: *If ''A'' and ''B'' are disjoint closed subsets whose union separates ''X'', then either ''A'' or ''B'' separates ''X''. *''X'' is unicoherent, meaning that if ''X'' is the union of two closed connected subsets, then their intersection is connected or empty. The theorem remains true with the weaker condition that ''A'' and ''B'' be separated.

References

* * * * García-Maynez, A. and Illanes, A. ‘A survey of multicoherence’, An. Inst. Autonoma Mexico 29 (1989) 17–67. * * Wilder, R. L. Topology of manifolds, AMS Colloquium Publications, Volume 32. American Mathematical Society, New York (1949). {{DEFAULTSORT:Phragmen-Brouwer theorem Theorems in topology Trees (topology)