In mathematics, a unicoherent space is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

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 entirely of open, connected sets. Background Throughout the history of topology, connectedness a ...

then it is called a

dendrite Dendrites (from Greek δένδρον ''déndron'', "tree"), also dendrons, are branched protoplasmic extensions of a nerve cell that propagate the electrochemical stimulation received from other neural cells to the cell body, or soma, of the n ...

. 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 Kong ...

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