Borsuk (1954) proved that dendroids have the
fixed-point property: Every continuous function from a dendroid to itself has a fixed point.[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.
[7]
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. This problem remains open.[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.[9][10]
A locally connected dendroid is called a
dendrite. A cone over the
Cantor set (called a
Cantor fan) is an example of a dendroid that is not a dendrite.[11]
^
abCharatonik, Janusz J. (1997), "The works of Bronisław Knaster (1893–1980) in continuum theory", Handbook of the history of general topology, Vol. 1, Dordrecht: Kluwer Acad. Publ., pp. 63–78,
MR1617581.
^
abBorsuk, K. (1954), "A theorem on fixed points", Bulletin de l'Académie polonaise des sciences. Classe troisième., 2: 17–20.
^Bellamy, David P. (1980), "A tree-like continuum without the fixed-point property", Houston J. Math., 6: 1–13,
MR0575909.
^Martínez-de-la-Vega, Veronica; Martínez-Montejano, Jorge M. (2011), "Open problems on dendroids", in Pearl, Elliott M. (ed.), Open Problems in Topology II, Elsevier, pp. 319–334,
ISBN9780080475295. See in particular p. 331.
^Minc, Piotr (2010), "An uncountable collection of dendroids mutually incomparable by continuous functions", Houston Journal of Mathematics, 36 (4): 1185–1205,
MR2753740. Previously announced in 2006.
^Islas, Carlos (2007), "An uncountable collection of mutually incomparable planar fans", Topology Proceedings, 31 (1): 151–161,
MR2363160.
^Charatonik, J.J.; Charatonik, W.J.; Miklos, S. (1990). "Confluent mappings of fans". Dissertationes Mathematicae. 301: 1–86.