= Solution
Statement (iii) is false. Let $\Omega$ be a thin open annulus about the unit circle, let $f(z)=1/z$, and split the circle into two proper closed arcs $K_1,K_2$ whose union is the whole circle. Each arc has connected complement, so <polynomial approximation of the reciprocal on a proper circular arc> gives uniform polynomial approximations to $f$ on each $K_j$ separately.
If polynomials $P_n$ converged uniformly to $1/z$ on $K_1\cup K_2$, then uniform convergence on the unit circle and the <Cauchy integral theorem> would give
$$
0=\lim_{n\to\infty}\int_{|z|=1}P_n(z)\,dz
=\int_{|z|=1}\frac{dz}{z}=2\pi i,
$$
a contradiction.
Back to article page
