Phragmen–Brouwer theorem explained

In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if X is a normal connected 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.
• Brown . R. . Antolín-Camarena . O. . Corrigendum to "Groupoids, the Phragmen–Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175–183 . 2014 . math.AT . 1404.0556.
• Wilder, R. L. Topology of manifolds, AMS Colloquium Publications, Volume 32. American Mathematical Society, New York (1949).