# 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 $({\tfrac {1}{2}},{\tfrac {1}{2}})\in \mathbb {R} ^{2}$ , 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 $X_{c}=\{(x,y)\in L(c):y\in \mathbb {Q} \}$ ; for all other points in $C$ let $X_{c}=\{(x,y)\in L(c):y\notin \mathbb {Q} \}$ ; the Knaster–Kuratowski fan is defined as $\bigcup _{c\in C}X_{c}$ equipped with the subspace topology inherited from the standard topology on $\mathbb {R} ^{2}$ .

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