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 $$\left(\tfrac1{2},\tfrac1{2}\right)\in\mathbb R^2$$, 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 $$p$$.