The Weierstrass approximation theorem states that any continuous function $ f : I \rightarrow \Bbb R $ on a closed, bounded, connected subset $ I \subseteq \Bbb R $ can be uniformly approximated by polynomials.
Can any continuous function $ \phi : J \rightarrow \Bbb C $ on a closed, bounded, connected subset $ J \subseteq \Bbb C $ be uniformly approximated by polynomials?
What I mean is, for which subsets $ J \subseteq \Bbb C $ can all functions be approximated uniformly by polynomials.
This question is an example sheet question that I had (already supervised on $-$ non-examinable) but supervisor wasn't sure what the question meant exactly. There are some basic sets, such as closed, real intervals, that it clearly holds for, but others (such as closed unit ball) that it does not hold for. Is anyone able to shed any light on the answer. (Not just a few counter-examples, but some explanation as to why it does / does not hold on certain set (eg, because connected complement / similar).)
Thanks very much!