In this question, $X$ and $Y$ are path-connected, locally simply connected spaces.

(a) Let $f: Y \rightarrow X$ be a continuous map, and $\widehat{X}$a path-connected covering space of $X$. State and prove a uniqueness statement for lifts of $f$ to $\widehat{X}$.

(b) Let $p: \widehat{X} \rightarrow X$ be a covering map. A covering transformation of $p$ is a homeomorphism $\phi: \widehat{X} \rightarrow \widehat{X}$such that $p \circ \phi=p$. For each integer $n \geqslant 3$, give an example of a space $X$ and an $n$-sheeted covering map $p_{n}: \widehat{X}_{n} \rightarrow X$ such that the only covering transformation of $p_{n}$ is the identity map. Justify your answer. [Hint: Take $X$ to be a wedge of two circles.]

(c) Is there a space $X$ and a 2-sheeted covering map $p_{2}: \widehat{X}_{2} \rightarrow X$ for which the only covering transformation of $p_{2}$ is the identity? Justify your answer briefly.