Exponential covering of an annulus by a vertical strip ($e^z:\{a<\mathrm{Re}z<b\}\twoheadrightarrow \{e^a<|w|<e^b\}$)

The complex complex exponential function has nonzero derivative and maps the strip onto the annulus, with fibers given by translations by $2\pi i$. It is locally conformal, but not injective. No biholomorphism from the full strip onto an annulus exists because the strip is simply connected space and the annulus has a loop of nonzero winding number. The complex exponential function is a covering map realizing that topological distinction.