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 ho ...

, 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. Biography and studies Kazimierz Kuratowski was born in Warsaw, ...

) is a specific connected

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, Lakȟ� ...

(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. 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 introduced by German mathematician Georg Cantor in 1883. T ...

, 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 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