Lemma 7

Let $n\geqslant 1$ be an integer and K a compact polynomially convex subset of $\mathbf{C}^n$. Then K admits a fundamental system of compact polynomially convex neighbourhoods.

There exists a compact polydisk (cf. VAR, R1, p. 24) Δ of $\mathbf{C}^n$ which is a neighbourhood of K. Since K is polynomially convex, there exists a family $(P_i)_{i\in I}$ of elements of $\mathbf{C}[X_1, . . . ,X_n]$, and a family $(M_i)_{i\in I}$ of positive real numbers such that K is the set of the $z\in$ Δ verifying $|P_i(z)|\leqslant M_i$ for every $i$ (lemma 3). For every finite subset J of I and every $\varepsilon >0$, let $K_{J,\varepsilon}$ be the set of the $z\in \Delta$ such that $|P_i(z)|\leqslant M_i+\varepsilon$ for $i\in J$. Then each set $K_{J,\varepsilon}$ is a compact polynomially convex neighbourhood of K (loc. cit.), and the intersection of the sets $K_{J,\varepsilon}$ is K. The sets $K_{J,\varepsilon}$ therefore form a fundamental system of polynomially convex neighbourhoods of K (TG, I, p. 60, th. 1).