| 词条 | Dendroid (topology) |
| 释义 |
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] {{harvtxt|Borsuk|1954}} proved that dendroids have the fixed-point property: Every continuous function from a dendroid to itself has a fixed point.[3] {{harvtxt|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 {{harvtxt|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 {{harvtxt|Minc|2010}} and {{harvtxt|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] References1. ^1 {{citation|last=Cook|first=H.|title=Continua: With the Houston Problem Book|volume=170|series=Lecture Notes in Pure and Applied Mathematics|publisher=CRC Press|year=1995|isbn=9780824796501|page=31|url=https://books.google.com/books?id=bVcZGMOA4AEC&pg=PA31}} {{DEFAULTSORT:Dendroid (Topology)}}{{Topology-stub}}2. ^1 {{citation | last = Charatonik | first = Janusz J. | contribution = The works of Bronisław Knaster (1893–1980) in continuum theory | location = Dordrecht | mr = 1617581 | pages = 63–78 | publisher = Kluwer Acad. Publ. | title = Handbook of the history of general topology, Vol. 1 | year = 1997}}. 3. ^1 {{citation|first=K.|last=Borsuk|title=A theorem on fixed points|year=1954|journal=Bulletin de l’Académie polonaise des sciences. Classe troisième.|volume=2|pages=17–20}}. 4. ^{{citation|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49124.pdf|first=A|last=Lelek|title=On plane dendroids and their end points in the classical sense|journal=Fund. Math.|year=1961|volume=49|pages=301–319}}. 5. ^{{citation | last = Cook | first = H. | journal = Fundamenta Mathematicae | mr = 0261558 | pages = 19–22 | title = Tree-likeness of dendroids and λ-dendroids | volume = 68 | year = 1970}}. 6. ^{{citation | last = Bing | first = R. H. | authorlink = R. H. Bing | journal = Duke Mathematical Journal | mr = 0043450 | pages = 653–663 | title = Snake-like continua | volume = 18 | year = 1951 | doi=10.1215/s0012-7094-51-01857-1}}. 7. ^{{citation | last = Bellamy | first = David P. | journal = Houston J. Math. | mr = 0575909 | pages = 1–13 | title = A tree-like continuum without the fixed-point property | volume = 6 | year = 1980}}. 8. ^{{citation|title=Open Problems in Topology II|editor-first=Elliott M.|editor-last=Pearl|publisher=Elsevier|year=2011|isbn=9780080475295|contribution=Open problems on dendroids|first1=Veronica|last1=Martínez-de-la-Vega|first2=Jorge M.|last2=Martínez-Montejano|pages=319–334}}. See in particular p. 331. 9. ^{{citation | last = Minc | first = Piotr | issue = 4 | journal = Houston Journal of Mathematics | mr = 2753740 | pages = 1185–1205 | title = An uncountable collection of dendroids mutually incomparable by continuous functions | volume = 36 | year = 2010}}. Previously announced in 2006. 10. ^{{citation | last = Islas | first = Carlos | issue = 1 | journal = Topology Proceedings | mr = 2363160 | pages = 151–161 | title = An uncountable collection of mutually incomparable planar fans | volume = 31 | year = 2007}}. 11. ^{{cite journal|first1=J.J.|last1=Charatonik|first2=W.J.|last2=Charatonik|first3=S.|last3=Miklos|title=Confluent mappings of fans|journal=Dissertationes Mathematicae|volume=301|year=1990|pages=1–86}} 2 : Continuum theory|Trees (topology) |
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