Knaster-Kuratowski fan

For any $a \in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$. Let $\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\mathcal{E}$ the endpoints of the removed intervals and $\mathcal{F} = \mathcal{C} \setminus \mathcal{E}$. Define $A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\}$ and $B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}$. This space is $X = A \cup B \subset \mathbb{R}^2$ with the subspace topology.

The subspace $X\setminus\{p\}$ obtained by removing the apex point is Punctured Knaster-Kuratowski fan.

Defined as counterexample #128 ("Cantor's Leaky Tent") in DOI 10.1007/978-1-4612-6290-9.