Consider a subset $K\subset \mathbb C^2$ consisting of pairs $(z,\bar z)$ such that $|z|=1$. Is there an easy way to see that continuous functions on $K$ can be uniformly approximated by polynomials in two variables?
The problem is not so trivial because if you take the set $L\subset \mathbb C^2$ of pairs $(z,0)$ with $|z|=1$ then $K$ is homeomorphic $L$, but it is clear that it is impossible to approximate the continuous function $\frac{1}{z}$ on $L$ by polynomials because of the Maximum Principle.
Thank you.