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.
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- 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