I was trying to understand that the image of analytic set needs not to be analytic.
The book provide a example that $\{|z|<1\}$ is analytic set and the map $\zeta \mapsto (\zeta^2-\zeta , \zeta^3 -\zeta)$ with image under this map, the point $(0,0)$ is not analytic.
I don't know how to prove it, I don't have a picture or intuition of the image of the open unit disc.