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][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]
References
- ↑ 1.0 1.1 Cook, H. (1995), Continua: With the Houston Problem Book, Lecture Notes in Pure and Applied Mathematics, 170, CRC Press, p. 31, ISBN 9780824796501, https://books.google.com/books?id=bVcZGMOA4AEC&pg=PA31
- ↑ 2.0 2.1 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.
- ↑ 3.0 3.1 Borsuk, K. (1954), "A theorem on fixed points", Bulletin de l'Académie polonaise des sciences. Classe troisième. 2: 17–20.
- ↑ Lelek, A (1961), "On plane dendroids and their end points in the classical sense", Fund. Math. 49 (3): 301–319, doi:10.4064/fm-49-3-301-319, http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49124.pdf.
- ↑ Cook, H. (1970), "Tree-likeness of dendroids and λ-dendroids", Fundamenta Mathematicae 68: 19–22, doi:10.4064/fm-68-1-19-22.
- ↑ "Snake-like continua", Duke Mathematical Journal 18 (3): 653–663, 1951, doi:10.1215/s0012-7094-51-01857-1.
- ↑ Bellamy, David P. (1980), "A tree-like continuum without the fixed-point property", Houston J. Math. 6: 1–13.
- ↑ Martínez-de-la-Vega, Veronica; Martínez-Montejano, Jorge M. (2011), "Open problems on dendroids", in Pearl, Elliott M., Open Problems in Topology II, Elsevier, pp. 319–334, ISBN 9780080475295. 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. Previously announced in 2006.
- ↑ Islas, Carlos (2007), "An uncountable collection of mutually incomparable planar fans", Topology Proceedings 31 (1): 151–161.
- ↑ Charatonik, J.J.; Charatonik, W.J.; Miklos, S. (1990). "Confluent mappings of fans". Dissertationes Mathematicae 301: 1–86.
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