Punctured Knaster-Kuratowski fan or Cantor's Teepee

Knaster-Kuratowski fan with the apex removed. Specifically: For any a[0,1]a \in [0,1], let L(a)L(a) be the line segment from (a,0)(a,0) to p=(12,12)p = (\frac{1}{2}, \frac{1}{2}). Let C\mathcal{C} be the middle-thirds Cantor set in the unit interval, E\mathcal{E} the endpoints of the removed intervals and F=CE\mathcal{F} = \mathcal{C} \setminus \mathcal{E}. Define A={(x,y)L(c)  cE,yQ}A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\} and B={(x,y)L(c)  cF,y∉Q}B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}. This space is X=(AB){p}R2X = (A \cup B) \setminus\{p\} \subset \mathbb{R}^2 with the subspace topology.

Defined as counterexample #129 ("Cantor's Teepee") in DOI 10.1007/978-1-4612-6290-9.