# Knaster–Kuratowski fan

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 $\bigcup_{c \in C} X_{c}$.
The fan itself is connected, but becomes totally disconnected upon the removal of $p$.