Different spokes must route through the hub

Consider ${\mathbb{R}}^2$ with the Euclidean metric d. Let $O=(0,0),P_0=(0,1)\in {\mathbb{R}}^2$, and for any integer $n\ge 1,P_n=(1/n,1)\in {\mathbb{R}}^2$. Let X = ⋃$(n\ge 0)L_n$, where $L_n$ is the line segment joining $P_n$ and O. Define $d_X:X\times X\to \mathbb{R}$ as follows: d_X(a, b) = d(a, b) when $a,b\in L_n$ for some n, and d_X(a, b) = d(a, O) + d(b, O) otherwise. Let $\tau$ be the smallest topology such that the sets $B_{(}a,\epsilon )=$ {$b\in X:d_X(a,b)<\epsilon$} are open in $\tau$ for all a ∈ X and $\epsilon >0$. Which of the following statements are true?