Continuous map from the ring on the unit circle Is there a surjective continuous map from the ring $r&lt;x^2+y^2&lt;1\,(0&lt;r&lt;1)$ on the unit circle $x^2+y^2&lt;1$ ?


It seems NO, but how can it be done ?


  Edit: what if we add the injectivity to the map ?
Yes. Let me denote $R = \sqrt{r}$, so the ring is defined by $R &lt; |p| &lt; 1$ where $p=(x,y)$. Divide the ring into an inner subring
$$R &lt; |p| \le (R+1)/2
$$
and an outer subring 
$$(R+1)/2 \le |p| &lt; 1
$$
Map the entire inner subring to the origin. Map the outer subring by pulling its inner boundary inward to the origin: for each angle $\theta$, the radial segment of the outer ring of angle $\theta$ connecting the circle of radius $(R+1)/2$ and the circle of radius $1$ is stretched by a constant factor over the radial segment of angle $\theta$ connecting the origin to the circle of radius $1$.
I find visual building of a map from an open ring to an open circle.

In the three dimension space imagine an open ring as a lateral surface of the cylinder. Now let pull the central line of the cylinder to the point. We have two cones with a common vertex and open boundaries. Finally, project (two-fold covering) these two cones to an open circle (two boundaries of cones map to the boundary of the circle).