On approximation by harmonic polynomials in the $C^1$-norm on compact sets in $\mathbf R^2$

It is proved that for an arbitrary compact set $X$ in $\mathbf R^2$ the following conditions are equivalent:
1) for every function $f\in C^1(\mathbf R^2)$, harmonic on $X^0$, and for any $\varepsilon>0$ a harmonic polynomial $p$ can be found such that
$$ \|f-p\|_X<\varepsilon,\qquad \|\nabla(f-p)\|_X<\varepsilon; $$
2) the set $\mathbf R^2\setminus X$ is connected