For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more generally, find the fundamental group $\pi_1(L_p)$ and a universal cover for $L_p$.
This question comes from an optional assignment sheet given after the completion of my introductory course on algebraic topology, and in particular I have done next to nothing on universal covers, so any help on how to even start approaching that part would be much appreciated (my lecturer didn't do any examples).
More importantly, I'm not even sure how to correctly interpret the quotient - should I be viewing $\mathbb{D}^2\sqcup\mathbb{S}^1$ sort of like a closed disc, and then imposing the equivalence relation on every point in this closed disc? This seems like it can't be the correct interpretation, since then $L_1$ is viewed as a closed disc without any equivalence relation, which is certainly not homeomorphic to the open disc? It seems that imposing a CW structure on $L_p$ would be useful in computing the fundamental groups, but again I haven't seen many examples of this, and I don't want to start on this until I've interpreted the question correctly!
If the interpretation is indeed ambiguous then I will ask my lecturer, but I think it's more likely I have misunderstood what's going on. If anyone can clear things up and explain how to proceed it would be much appreciated.