Knaster–Kuratowski fan explained

In topology, a branch of mathematics, the Knaster - Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point makes it totally disconnected. It is also known as Cantor's leaky tent or Cantor's teepee (after Georg Cantor), depending on the presence or absence of the apex.

Let

C be the Cantor set, let p be the point \left(\tfrac 1 { 2 },\tfrac 1 { 2 }\right)\in 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 _{c} = \{ (x,y) \in L(c) : y \in Q \} ; for all other points in C let X _{c} = \{ (x,y) \in L(c) : y \notin Q \} ; the Knaster - Kuratowski fan is defined as c up _{c} X _{c} equipped with the subspace topology inherited from the standard topology on R ^{2} .

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

p .

See also

• Antoine's necklace