(a) Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is continuous with $\gamma(0)=\gamma(1)$ and $\gamma(t) \neq 0$ for all $t \in[0,1]$. Show that if $\gamma(0)=|\gamma(0)| \exp \left(i \theta_{0}\right)$ (with $\theta_{0}$ real) we can define a continuous function $\theta:[0,1] \rightarrow \mathbb{R}$ such that $\theta(0)=\theta_{0}$ and $\gamma(t)=|\gamma(t)| \exp (i \theta(t))$. Hence define the winding number $w(\gamma)=w(0, \gamma)$ of $\gamma$ around 0 .

(b) Show that $w(\gamma)$ can take any integer value.

(c) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition, and $\left(\gamma_{1} \times \gamma_{2}\right)(t)=\gamma_{1}(t) \gamma_{2}(t)$, show that

$$w\left(\gamma_{1} \times \gamma_{2}\right)=w\left(\gamma_{1}\right)+w\left(\gamma_{2}\right)$$

(d) If $\gamma_{1}$ and $\gamma_{2}$ satisfy the requirements of the definition and $\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)\right|$ for all $t \in[0,1]$, show that

$$w\left(\gamma_{1}\right)=w\left(\gamma_{2}\right)$$

(e) State and prove a theorem that says that winding number is unchanged under an appropriate homotopy.