topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster 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. He worked as a professor at the University of Warsaw and at the Ma ...

) is a specific connected

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

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 (after

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...

), 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-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...

. Let

C

be the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

, 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)

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 In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...

inherited from the standard topology on

\mathbb^2

. The fan itself is connected, but becomes totally disconnected upon the removal of

p

References

* * Topological spaces {{topology-stub

See also

References