Define what it means for a map $p: \widetilde{X} \rightarrow X$ to be a covering space. State the homotopy lifting lemma.

Let $p:\left(\tilde{X}, \tilde{x}_{0}\right) \rightarrow\left(X, x_{0}\right)$ be a based covering space and let $f:\left(Y, y_{0}\right) \rightarrow\left(X, x_{0}\right)$ be a based map from a path-connected and locally path-connected space. Show that there is a based lift $\tilde{f}:\left(Y, y_{0}\right) \rightarrow\left(\tilde{X}, \tilde{x}_{0}\right)$ of $f$ if and only if $f_{*}\left(\pi_{1}\left(Y, y_{0}\right)\right) \subseteq p_{*}\left(\pi_{1}\left(\widetilde{X}, \tilde{x}_{0}\right)\right)$.