$p$ is polynomial, set bounded open set with at most $n$ components Assume $p$ is a non constant polynomial of degree $n$. Prove that the set $\{z:|(p(z))| \lt 1\}$ is a bounded open set with at-most $n$ connected components. Give example to show number of components can be less than $n$.
thanks.
EDIT:Thanks,I meant connected components.

Hints:


*

*For boundedness: Show that $|p(z)| \to \infty$ as $|z|\to \infty$

*For openness: The preimage $p^{-1}(A)$ of an open set $A$ under a continuous function $p$ is again open. Polynomials are continuous, so just write your set as a preimage.

*For the connected components: Recall fundamental theorem of algebra. What may these components have to do with polynomial roots? How many of them can a polynomial have? The example for fewer components is straightforward, just think of it geometrically (draw a curve). 
