Unicoherent space
In mathematics, a unicoherent space is a topological space [math]\displaystyle{ X }[/math] that is connected and in which the following property holds: For any closed, connected [math]\displaystyle{ A, B \subset X }[/math] with [math]\displaystyle{ X=A \cup B }[/math], the intersection [math]\displaystyle{ A \cap B }[/math] is connected.
For example, any closed interval on the real line is unicoherent, but a circle is not.
If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and arcwise connected, then it is called a dendroid. If in addition it is locally connected then it is called a dendrite. The Phragmen–Brouwer theorem states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space.
References
- Charatonik, Janusz J. (2003). "Unicoherence and Multicoherence". Encyclopedia of General Topology. pp. 331–333. doi:10.1016/B978-044450355-8/50088-X. ISBN 9780444503558. https://books.google.com/books?id=JWyoCRkLFAkC&pg=PA331.
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