Dendroid (topology)

In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of X is unicoherent), arcwise connected, and forms a continuum.

[1] The term dendroid was introduced by Bronisław Knaster lecturing at the University of Wrocław,[2] although these spaces were studied earlier by Karol Borsuk and others.

[3] Cook (1970) proved that every dendroid is tree-like, meaning that it has arbitrarily fine open covers whose nerve is a tree.

[1][5] The more general question of whether every tree-like continuum has the fixed-point property, posed by Bing (1951),[6] was solved in the negative by David P. Bellamy, who gave an example of a tree-like continuum without the fixed-point property.

[2][8] 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 onto any other dendroid in the collection, was solved by Minc (2010) and Islas (2007), who gave an example of such a family.