Winding number of a continuous closed path

If $\gamma :[0,1]\to \mathbb{C}\setminus \{0\}$ is continuous and closed, choose a continuous lift $\theta$ such thatThen $\mathrm{wind}(\gamma ,0)=\theta (1)-\theta (0)\in \mathbb{Z}$. This agrees with the contour-integral definition when the path is piecewise smooth.

On a locally path-connected space, on each path component choose a value of a complex logarithm of $f$ at a base point. Continue it along paths using local logarithms. Independence of the path is exactly vanishing of the winding number of a continuous closed path for all loops. This constructs a continuous function $g$ with $f=e^g$. The implication from a logarithm to zero winding requires no local connectivity. For applications to arbitrary planar compact sets, apply the criterion on polygonal neighborhoods rather than assuming the compact set is locally path-connected.