Unicoherent space

In mathematics, a unicoherent space is a topological space ${\displaystyle X}$ that is connected and in which the following property holds:

For any closed, connected ${\displaystyle A,B\subset X}$ with ${\displaystyle X=A\cup B}$, the intersection ${\displaystyle A\cap B}$ 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.

External links

• Insall, Matt. "Unicoherent Space". MathWorld.