The winding number enjoys the following properties:

1. For any closed curve $\gamma$, the winding number $n_{\gamma }(0)\in \mathbb{Z}$
2. The winding number is homotopy invariant (we can get a continuous integer valued function).
3. The winding number is invariant under a continuous orientation preserving reparameterization (it doesn't matter how fast we go).
4. $n_{\gamma }(a)$ is locally constant with respect to $a$. That is, if $a$ changes continuously not meeting the curve, then the winding number does not change. In particular, for $a$ in any connected component of $\mathbb{C}-im(\gamma )$, the value of the winding number does not change.