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In topology, a branch of mathematics, the Knaster–Kuratowski fan 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 , depending on the presence or absence of the apex.
Let C {\displaystyle C} be the Cantor set, let p {\displaystyle p} be the point ∈ R 2 {\displaystyle \left\in \mathbb {R} ^{2}} , and let L {\displaystyle L} , for c ∈ C {\displaystyle c\in C} , denote the line segment connecting {\displaystyle } 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 = { ∈ L : y ∈ Q } {\displaystyle X_{c}=\{\in L:y\in \mathbb {Q} \}} ; for all other points in C {\displaystyle C} let X c = { ∈ L : y ∉ Q } {\displaystyle X_{c}=\{\in L: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}.