Can the winding number be a non-integer? The formal definition of a winding number: > For a continuous loop $\gamma\colon[\alpha,\beta]\to\mathbb{C}\setminus\\{a\\}$ which doesn't pass through a point $a$, one has the function $\theta(t)=\text{arg}(\gamma(t)-a)\in\mathbb{R}/2\pi\mathbb{Z}$. By the lifting lemma, there exists a continuous $\tilde{\theta}\colon[\alpha,\beta]\to\mathbb{R}$, such that $[\tilde{\theta}(t)]=\theta(t)$, and the winding number of $\gamma$ around $a$ is then defined as $$n(\gamma,a)=\frac{\tilde{\theta}(\beta)-\tilde{\theta}(\alpha)}{2\pi}.$$ **Can't the winding number be a non-integer? How can we ensure that the winding number is an integer?**

Since $\gamma$ is a loop, $\gamma(\alpha) = \gamma(\beta)$, and so $$\theta(\gamma(\beta)) - \theta(\gamma(\alpha)) = 0 \in \mathbb{R} / 2 \pi \mathbb{Z}.$$ Thus, for any lift $\tilde{\theta}$ of $\theta$, $$\tilde{\theta}(\gamma(\beta)) - \tilde{\theta}(\gamma(\alpha)) \in 2 \pi \mathbb{Z},$$ and so $$n(\gamma, a) \in \mathbb{Z}.$$