Relation between a map and its lifting

[yaa09d]

I have the following question:
Let $\mathbb{D}$ denote the unit disk.
Let $f:X_1 \longrightarrow X_2$ be a continuous mapping between Riemann Surfaces.
Let $ \pi_1 : \mathbb{D} \longrightarrow X_1$ , and $ \pi_2 : \mathbb{D} \longrightarrow X_2$ be the universal covering spaces of $X_1$ and $X_2$, respectively. A lifting of $f$ is a continuous map $ \tilde{f}: \mathbb{D}\longrightarrow \mathbb{D}$ such that $f\circ \pi_1=\pi_2\circ \tilde{f}.$

The question is to show if $f$ is homeomorphism, then so is $\tilde{f},$ or to give a counterexample.

Any help will be appreciated.

Thank you.

 

[pat_connell]

Answer: The statement is true. If $f$ is a homeomorphism, then it is a bijection and hence has an inverse $f^{-1}$. We can then construct $\tilde{f}^{-1}$, the lifting of $f^{-1}$, which is also continuous. Since $f$ and $f^{-1}$ are mutually inverse homeomorphisms, so are $\tilde{f}$ and $\tilde{f}^{-1}$. Thus, $\tilde{f}$ is a homeomorphism.