Phragmen–Brouwer theorem

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

R.F. Dickman jr (1984), "A Strong Form of the Phragmen–Brouwer Theorem", Proceedings of the American Mathematical Society, 90 (2): 333–337, doi:10.2307/2045367, JSTOR 2045367
Hunt, J.H.V. (1974), "The Phragmen–Brouwer theorem for separated sets", Bol. Soc. Mat. Mex., Series II, 19: 26–35, Zbl 0337.54021
Wilson, W. A. (1930), "On the Phragmén–Brouwer theorem", Bulletin of the American Mathematical Society, 36 (2): 111–114, doi:10.1090/S0002-9904-1930-04901-0, ISSN 0002-9904, MR 1561900
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". arXiv:1404.0556.
Wilder, R. L. Topology of manifolds, AMS Colloquium Publications, Volume 32. American Mathematical Society, New York (1949).