Here is a picture of a helix H and an annulus A. You should think of the helix as being infinite in both directions, although we only have space to show a few loops. Vertical projection gives a map p : H -> A. We use the points marked in red as basepoints for H and A.

Let S be a sector in A as shown at the bottom. Then the preimage p

Suppose we have a closed curve u : S

The section of the curve that we have already lifted is shown in blue; we now try to extend this lift to the next section of the curve, which is shown in red at the bottom. There are again infinitely many ways to lift the red section, of which we have shown three. Only one of these (shown in red) joins up with our lift of the first section of the curve; we choose this one.

The section of the curve that we have already lifted is shown in blue; we now try to extend this lift to the next section of the curve, which is shown in red at the bottom. There are again infinitely many ways to lift the red section, of which we have shown three. Only one of these (shown in red) joins up with our lift of the first section of the curve; we choose this one.

The section of the curve that we have already lifted is shown in blue; we now try to extend this lift to the next section of the curve, which is shown in red at the bottom. There are again infinitely many ways to lift the red section, of which we have shown three. Only one of these (shown in red) joins up with our lift of the previous sections of the curve; we choose this one.

The section of the curve that we have already lifted is shown in blue; we now try to extend this lift to the last section of the curve, which is shown in red at the bottom. There are again infinitely many ways to lift the red section, of which we have shown three. Only one of these (shown in red) joins up with our lift of the previous sections of the curve; we choose this one.

We now have a lift of the whole curve to H, shown in blue. However, the lifted curve does not close up; it ends in a different place to where it started. The gap between them is shown in red. The end is one sheet above the beginning, which mean that the winding number of the original curve is 1.