Let $K_1\subset K_2 \subset \mathbb{C}$ be compact. Let $h(z)$ be a function defined by
\begin{align*} h(z) = \left\{ \begin{matrix} 1 & \text{if }z\in K_1\\ 0 & \qquad\text{ if }z\in K_2\setminus K_1 \end{matrix} \right. \end{align*}
Can we say $h(z)$ can be approximated uniformly by polynomials of $z$ for arbitrary ? If not, what about the case when $K_1$ and $K_2$ have no interior?
There is the obvious obstruction that, if $K_2$ is connected and $\emptyset\subsetneq K_1\subsetneq K_2$, then $h$ is not continuous on $K_2$, and therefore it cannot be approximated uniformly by continuous functions. Similarly, this is the case as soon as $K_1$ is not union of connected components of $K_2$.