Suppose $p:Y \rightarrow X$ is the universal covering map of $X$. Given a contiunuous $f: X \rightarrow X$ then a well known theorem for existence of lifts states that there exist a continuous lift $\tilde f : Y \rightarrow Y \text{ with } p \circ \tilde f = f \circ p.$ If we additionally suppose f is a homeomorphism. Then I think $\tilde f$ is a homeomorphism, too. I tried to prove that but failed. My idea was to take a lift of $f^{-1}$ and compose it with $\tilde f$ in order to obtain a lift of the identity. But that´s not really helping here. Anybody knows how to prove or disprove that?
Thanks in advance