Knaster–Kuratowski fan

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 ( 1 2 , 1 2 ) ∈ R 2 , and let L ( c ) , for c ∈ C , denote the line segment connecting ( c , 0 ) to p . If c ∈ C is an endpoint of an interval deleted in the Cantor set, let X c = { ( x , y ) ∈ L ( c ) : y ∈ Q } ; for all other points in C let X c = { ( x , y ) ∈ L ( c ) : y ∉ Q } ; the Knaster–Kuratowski fan is defined as ⋃ c ∈ 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 .