In mathematics, a unicoherent space is a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

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 In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if ev ...

then it is called a

dendrite A dendrite (from Ancient Greek language, Greek δένδρον ''déndron'', "tree") or dendron is a branched cytoplasmic process that extends from a nerve cell that propagates the neurotransmission, electrochemical stimulation received from oth ...

. The

Phragmen–Brouwer theorem In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if ''X'' is a normal connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kon ...

states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space.

References

External links

* General topology Trees (topology) {{topology-stub