On non-Hausdorff manifolds

In this long note, we investigate various purely topological aspects of non-Hausdorff manifolds (NH-manifolds for short). Our emphasis is on manifolds which exhibit homogeneity or weakenings thereof, in particular being everywhere non-Hausdorff. Homogeneous NH-manifolds and everywhere non-Hausdorff manifolds are respectively called HNH- and ENH-manifolds. We write $N{H}_X(x)$ for the subset of points of a space $X$ which cannot be separated of $x$ by open sets. The topics covered in this note are the following. – General (basic) properties of manifolds and their quasi-compact or quasi-countably compact subspaces. – Covering properties implying the Hausdorffness of (weakly) homogeneous manifolds. – (Non-)existence of hereditarily separable ENH-manifolds (under set theoretic hypotheses). – Non-existence of a quasi-countably compact ENH-manifold. – Properties of NH-manifolds which imply that $N{H}_M(x)$ is discrete, or at least simple''. -- Constructions of HNH-manifolds such that $NH(x)$ is non-homogeneous, for instance a countable union of closed intervals and $n$-torii. -- Constructions of NH-manifolds $M$ with a point $x$ such that $NH_M(x)$ is homeomorphic to various complicated’’ spaces, in particular in dimension $1$ and $2$. We use elementary (or at least well known) methods of general or set theoretic topology, with a little bit of conformal theory and dynamical systems (flows) in some constructions. Many pictures are given to illustrate the constructions, and the proofs are rather detailed, which is the main reason for the length of this note.