Let $T=S^{1} \times S^{1}$ be the 2-dimensional torus, and let $X$ be constructed from $T$ by removing a small open disc.

(a) Show that $X$ is homotopy equivalent to $S^{1} \vee S^{1}$.

(b) Show that the universal cover of $X$ is homotopy equivalent to a tree.

(c) Exhibit (finite) cell complexes $X, Y$, such that $X$ and $Y$ are not homotopy equivalent but their universal covers $\widetilde{X}, \widetilde{Y}$are.

[State carefully any results from the course that you use.]