In mathematics, a dendroid is a type of

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 po ...

, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of ''X'' is unicoherent),

arcwise connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...

, and forms a continuum. The term dendroid was introduced by Bronisław Knaster lecturing at the

University of Wrocław , ''Schlesische Friedrich-Wilhelms-Universität zu Breslau'' (before 1945) , free_label = Specialty programs , free = , colors = Blue , website uni.wroc.pl The University of Wrocław ( pl, Uniwersytet Wrocławski, U ...

,. although these spaces were studied earlier by

Karol Borsuk Karol Borsuk (May 8, 1905 – January 24, 1982) was a Polish mathematician. His main interest was topology, while he obtained significant results also in functional analysis. Borsuk introduced the theory of ''absolute retracts'' (ARs) and '' abs ...

and others.. proved that dendroids have the

fixed-point property A mathematics, mathematical object ''X'' has the fixed-point property if every suitably well-behaved mapping (mathematics), mapping from ''X'' to itself has a fixed point (mathematics), fixed point. The term is most commonly used to describe topolog ...

: Every continuous function from a dendroid to itself has a fixed point. proved that every dendroid is ''tree-like'', meaning that it has arbitrarily fine open covers whose

nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the ...

is a tree. The more general question of whether every tree-like continuum has the fixed-point property, posed by , was solved in the negative by David P. Bellamy, who gave an example of a tree-like continuum without the fixed-point property. In Knaster's original publication on dendroids, in 1961, he posed the problem of characterizing the dendroids which can be embedded into the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

. This problem remains open. Another problem posed in the same year by Knaster, on the existence of an uncountable collection of dendroids with the property that no dendroid in the collection has a continuous

surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

onto any other dendroid in the collection, was solved by and , who gave an example of such a family. A locally connected dendroid 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 ...

. A cone over the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...

(called a Cantor fan) is an example of a dendroid that is not a dendrite.

References

Continuum theory Trees (topology) {{Topology-stub