Define what it means for $p: \widetilde{X} \rightarrow X$ to be a covering map, and what it means to say that $p$ is a universal cover.

Let $p: \tilde{X} \rightarrow X$ be a universal cover, $A \subset X$ be a locally path connected subspace, and $\tilde{A} \subset p^{-1}(A)$ be a path component containing a point $\tilde{a}_{0}$ with $p\left(\tilde{a}_{0}\right)=a_{0}$. Show that the restriction $\left.p\right|_{\tilde{A}}: \widetilde{A} \rightarrow A$ is a covering map, and that under the Galois correspondence it corresponds to the subgroup

$$\operatorname{Ker}\left(\pi_{1}\left(A, a_{0}\right) \rightarrow \pi_{1}\left(X, a_{0}\right)\right)$$

of $\pi_{1}\left(A, a_{0}\right)$.