Uncountable fundamental group.

Question

I was trying to understand what is an example of a topological space that has an uncountable fundamental group. I was reading this answer but I don't understand what $L_q \equiv 0$ and $L_q \equiv 1$ mean.

Answer 1

$L_q \equiv 0$ means that the loop $L_p$ is null-homotopic in $\mathbb{R}^2-(r,r)$. $L_q \equiv 1$ is certainly an infelicitous wording. However, it means that the loop is the canonical generator of $\pi_1(\mathbb{R}^2-(r,r)) \approx \mathbb Z$.

If the loops $L_{q_1}$ and $L_{q_2}$ would be homotopic in $\mathbb{R}^2- \mathbb Q^2$, then also in the bigger space $\mathbb{R}^2-(r,r)$. This is not true in the given situation.