Let $𝕋$ be a complex one-dimensional torus. We prove that all subsets of $𝕋$ with finitely many boundary components (none of them being points) embed properly into ${ℂ}^2$. We also show that the algebras of analytic functions on certain countably connected subsets of closed Riemann surfaces are doubly generated.

Due to a technical error, part of a sentence was omitted on the top of page 8. The first line should read: “where $f_p^k$, $p=a_l$ or $b_l$, means the number of folds of the covering $({\delta }_k^{\text{'}\text{'}},T|,{\Delta }_l^{\text{'}\text{'}})$ ending at p, i.e. covering a neighbourhood of p in $a_l{b}_l$ without covering p itself”.

Nous construisons une famille de surfaces de Riemann hyperelliptiques, de genre variable, munies de fonctions méromorphes de degré deux et d’indice un, ce qui apporte une réponse positive à une conjecture de S. Montiel et A. Ros.