Knaster–Kuratowski fan
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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians
Bronisław Knaster Bronisław Knaster (22 May 1893 – 3 November 1980) was a Polish mathematician; from 1939 a university professor in Lwów and from 1945 in Wrocław. He is known for his work in point-set topology and in particular for his discoveries in 1922 of ...
and
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Biography and studies Kazimierz Kuratowski was born in Warsaw, (t ...
) is a specific
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
with the property that the removal of a single point makes it
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
. It is also known as Cantor's leaky tent or Cantor's
teepee A tipi , often called a lodge in English, is a conical tent, historically made of animal hides or pelts, and in more recent generations of canvas, stretched on a framework of wooden poles. The word is Siouan, and in use in Dakhótiyapi, Lakȟó ...
(after
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
), depending on the presence or absence of the
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, ...
. Let C be the Cantor set, let p be the point \left(\tfrac1,\tfrac1\right)\in\mathbb R^2, and let L(c), for c \in C, denote the line segment connecting (c,0) to p. If c \in C is an endpoint of an interval deleted in the Cantor set, let X_ = \; for all other points in C let X_ = \; the Knaster–Kuratowski fan is defined as \bigcup_ X_ equipped with the subspace topology inherited from the standard topology on \mathbb^2. The fan itself is connected, but becomes totally disconnected upon the removal of p.


See also

*
Antoine's necklace In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeom ...


References

* * Topological spaces {{topology-stub