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 {\displaystyle C} be the Cantor set, let p {\displaystyle p} be the point ( 1 2 , 1 2 ) ∈ R 2 {\displaystyle \left({\tfrac {1}{2}},{\tfrac {1}{2}}\right)\in \mathbb {R} ^{2}} , and let L ( c ) {\displaystyle L(c)} , for c ∈ C {\displaystyle c\in C} , denote the line segment connecting ( c , 0 ) {\displaystyle (c,0)} to p {\displaystyle p} . If c ∈ C {\displaystyle c\in C} is an endpoint of an interval deleted in the Cantor set, let X c = { ( x , y ) ∈ L ( c ) : y ∈ Q } {\displaystyle X_{c}=\{(x,y)\in L(c):y\in \mathbb {Q} \}} ; for all other points in C {\displaystyle C} let X c = { ( x , y ) ∈ L ( c ) : y ∉ Q } {\displaystyle X_{c}=\{(x,y)\in L(c):y\notin \mathbb {Q} \}} ; the Knaster–Kuratowski fan is defined as ⋃ c ∈ C X c {\displaystyle \bigcup _{c\in C}X_{c}} equipped with the subspace topology inherited from the standard topology on R 2 {\displaystyle \mathbb {R} ^{2}} .

The fan itself is connected, but becomes totally disconnected upon the removal of p {\displaystyle p} .