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 a fixed-point theorem: 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 question of whether more generally every tree-like continuum has a fixed-point theorem, posed by Bing (1951) ^{[6]} has been solved by David P. Bellamy ^{[7]}
In Knaster's original publication on dendroids, in 1961, he posed the problem of characterizing the dendroids that can be embedded into the Euclidean plane, which also 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).^{[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
- ^ ^{a} ^{b} Cook, H. (1995), Continua: With the Houston Problem Book, Lecture Notes in Pure and Applied Mathematics, 170, CRC Press, p. 31, ISBN 9780824796501
- ^ ^{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.
- ^ ^{a} ^{b} 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" (PDF), Fund. Math., 49: 301–319.
- ^ Cook, H. (1970), "Tree-likeness of dendroids and λ-dendroids", Fundamenta Mathematicae, 68: 19–22, MR 0261558.
- ^ Bing, R. H. (1951), "Snake-like continua", Duke Mathematical Journal, 18: 653–663, MR 0043450, 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, MR 0575909.
- ^ 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, MR 2753740. Previously announced in 2006.
- ^ Islas, Carlos (2007), "An uncountable collection of mutually incomparable planar fans", Topology Proceedings, 31 (1): 151–161, MR 2363160.
- ^ Charatonik, J.J.; Charatonik, W.J.; Miklos, S. (1990). "Confluent mappings of fans". Dissertationes Mathematicae. 301: 1–86.
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