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, aft ...
and
Luitzen Egbertus Jan Brouwer, states that if ''X'' is a
normal connected locally connected topological space
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness a ...
, 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 In mathematics, a unicoherent space is a topological space X that is connected space , connected and in which the following property holds:
For any closed, connected A, B \subset X with X=A \cup B, the intersection A \cap B is connected.
For exam ...
, 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).
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Theorems in topology
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