Lemma 1

Let X be a compact subset of the plane and let O be a bounded connected component of $\mathbf{C}-$ X. The boundary of O is contained in X.

The closure of the set O in $\mathbf{C}-$ X is equal to $\overline{O}\cap (\mathbf{C}-$ X) where $\overline{O}$ is its closure in $\mathbf{C}$. Since O is a connected component of $\mathbf{C}-$ X, we therefore have $\overline{O}\cap (\mathbf{C}-$ X) = O, which proves that $\overline{O}-O\subset X$.

Let X be a compact subset of $\mathbf{C}$. Let $O_{\infty}$ be the unbounded connected component of $\mathbf{C}-$X, and let $(O_i)_{i\in I}$ be the family of bounded connected components of $\mathbf{C}-$ X, the subsets $O_i$ being pairwise distinct.

Let E be a subset of $\mathbf{C}-$ X. We denote by $R_E(X)$ the closure in $\mathscr{C}(X)$ of the set of functions $f|X$, where $f$ is a rational function on $\mathbf{C}$ all of whose poles belong to E. The algebra $R_E(X)$ is a unital Banach subalgebra of $\mathscr{C}(X)$ which separates the points of X. Let $z$ be the identity function on X. The full closed subalgebra of $R_E(X)$ generated by $z$ is equal to $R_E(X)$ (Lemma 2 of I, p. 6). The elements of $R_E(X)$ are holomorphic in the interior of X.

In particular, one has $R_{\emptyset}(X) = P(X)$. We write $R(X) = R_{\mathbf{C}-X}(X)$. Let I(E) be the set of elements $i\in I$ such that $E\cap O_i=\emptyset$, and

The set $X_E$ is compact, being bounded and closed, its complement in $\mathbf{C}$ being the union of the open set $O_{\infty}$ and of the open sets $O_i$ which meet E.