Unicoherent space explained

X that is 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 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

• Book: 10.1016/B978-044450355-8/50088-X . Unicoherence and Multicoherence . [{{Google books|JWyoCRkLFAkC|page=331|plainurl=yes}} Encyclopedia of General Topology ]. 2003 . Charatonik . Janusz J. . 331–333 . 9780444503558.