Define the notions of covering projection and of locally path-connected space. Show that a locally path-connected space is path-connected if it is connected.

Suppose $f: Y \rightarrow X$ and $g: Z \rightarrow X$ are continuous maps, the space $Y$ is connected and locally path-connected and that $g$ is a covering projection. Suppose also that we are given base-points $x_{0}, y_{0}, z_{0}$ satisfying $f\left(y_{0}\right)=x_{0}=g\left(z_{0}\right)$. Show that there is a continuous $\tilde{f}: Y \rightarrow Z$ satisfying $\tilde{f}\left(y_{0}\right)=z_{0}$ and $g \tilde{f}=f$ if and only if the image of $f_{*}: \Pi_{1}\left(Y, y_{0}\right) \rightarrow \Pi_{1}\left(X, x_{0}\right)$ is contained in that of $g_{*}: \Pi_{1}\left(Z, z_{0}\right) \rightarrow \Pi_{1}\left(X, x_{0}\right)$. [You may assume the path-lifting and homotopy-lifting properties of covering projections.]

Now suppose $X$ is locally path-connected, and both $f: Y \rightarrow X$ and $g: Z \rightarrow X$ are covering projections with connected domains. Show that $Y$ and $Z$ are homeomorphic as spaces over $X$ if and only if the images of their fundamental groups under $f_{*}$ and $g_{*}$ are conjugate subgroups of $\Pi_{1}\left(X, x_{0}\right)$.