Proposition 13

Let $n\geqslant 1$ be an integer. Let $K\subset \mathbf{C}^n$ be a closed subset and V its polynomially convex hull.

a) Every bounded connected component of $\mathbf{C}^n-$ K is contained in V;

b) If $n= 1$, and if K is compact, then V is the union $\widehat{K}$ of K and of the bounded connected components of $\mathbf{C}-$ K.

Since $K\subset \mathbf{C}^n$ is closed, its complement $\mathbf{C}^n-$ K is open, hence locally connected, so that each connected component of $\mathbf{C}^n-$ K is open. The maximum principle (VAR, R1, p. 29, 3.3.7) then shows that every bounded connected component of $\mathbf{C}^n-$ K is contained in V, which proves assertion a).

Now suppose $n= 1$ and K compact. Let us denote by $\widehat{K}$ the union of K and of the bounded connected components of $\mathbf{C}-$ K, so that $\widehat{K}\subset$ V by what precedes. The set $\widehat{K}$ is compact (Lemma 5). Let A be the commutative unital Banach algebra $\mathscr{C}(\widehat{K})$. Let $x\in A$ be the identity function of $\widehat{K}$ and B the closed unital subalgebra of A generated by $x$. We have Sp$_A(x) =\widehat{K}$ (Example 3 of I, p. 17), hence $\mathbf{C}-$ Sp$_A(x)$ is connected. Consequently we have Sp$_B(x) =$ Sp$_A(x)$ (Cor. of Prop. 6 of I, p. 28). Since Sp$_B(x)$ is polynomially convex by Lemma 4, this shows that $\widehat{K}$ is polynomially convex and therefore that $V\subset \widehat{K}$.

The second part of the proposition does not extend to the case $n\geqslant 2$ (cf. Exercise 23 of I, p. 170).