One dimensional non-Hausdorff manifolds and foliations of the plane

Abstract

The original article titled "Vari\'et\'es (non s\'epar\'ees) \`a une dimension et structures feuillet\'ees du plan" was published in 1957 in French in L'Enseignement Math\'ematique. It establishes a beautiful connection between foliations of the plane and non-Hausdorff $1$-dimensional manifolds arising naturally as leaf spaces of the foliations. Since its appearance, this theory has paved the way for several results concerning dynamical systems and foliations of the plane and $2$-manifolds. Haefliger and Reeb's article inspires many results in the theory of foliation of $3$-manifolds as well: we refer the interested reader to Danny Calegari's book on the topic. Haefliger and Reeb's article also has been applied to areas outside topological dynamics. This article has been referenced in $43$ papers to-date and has helped build many nice results. In the literature, the main theorem of this article has been commonly referred to as Haefliger-Reeb theory or "a classical result by Haefliger and Reeb". However, to the best of our knowledge, no English version of this article is currently available. We have kept the article entirely unchanged except to add figures and correct a few typographical errors. We hope that this translation will be useful for the broader audience.