Dendroid (topology)

A dendrite such as this Julia set is a (locally connected) dendroid.

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][4]

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]

  1. ^ a b Cook, H. (1995), Continua: With the Houston Problem Book, Lecture Notes in Pure and Applied Mathematics, vol. 170, CRC Press, p. 31, ISBN 9780824796501
  2. ^ a b Charatonik, 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, MR 1617581.
  3. ^ a b Borsuk, K. (1954), "A theorem on fixed points", Bulletin de l'Académie polonaise des sciences. Classe troisième., 2: 17–20.
  4. ^ Lelek, A (1961), "On plane dendroids and their end points in the classical sense" (PDF), Fund. Math., 49 (3): 301–319, doi:10.4064/fm-49-3-301-319.
  5. ^ Cook, H. (1970), "Tree-likeness of dendroids and λ-dendroids", Fundamenta Mathematicae, 68: 19–22, doi:10.4064/fm-68-1-19-22, MR 0261558.
  6. ^ Bing, R. H. (1951), "Snake-like continua", Duke Mathematical Journal, 18 (3): 653–663, doi:10.1215/s0012-7094-51-01857-1, MR 0043450.
  7. ^ Bellamy, David P. (1980), "A tree-like continuum without the fixed-point property", Houston J. Math., 6: 1–13, MR 0575909.
  8. ^ 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, ISBN 9780080475295. See in particular p. 331.
  9. ^ Minc, Piotr (2010), "An uncountable collection of dendroids mutually incomparable by continuous functions", Houston Journal of Mathematics, 36 (4): 1185–1205, MR 2753740. Previously announced in 2006.
  10. ^ Islas, Carlos (2007), "An uncountable collection of mutually incomparable planar fans", Topology Proceedings, 31 (1): 151–161, MR 2363160.
  11. ^ Charatonik, J.J.; Charatonik, W.J.; Miklos, S. (1990). "Confluent mappings of fans". Dissertationes Mathematicae. 301: 1–86.