Covering space from $D^{2} \rightarrow S^{1}$? Is there a covering space $p : D^2 \rightarrow S^1$? I'm not sure how to go about solving this problem. I considered maps such as $z \rightarrow \frac{z+a}{|z+a|}$, but I'm not sure how to show where or not this is satisfies all of the conditions....

Well, $D^2$ is simply connected so if there is a covering map $p$ then $p:D^2\to S^1$ is a universal cover of $S^1$. If you are aware of the following two facts then you have your answer:
$1)$ Universal covers are unique up to covering space isomorphism (which in particular is a homeomorphism).
$2)$ $\theta\mapsto e^{2\pi i\theta}:\Bbb R\to S^1$ is a universal cover of $S^1$.
