On non-Hausdorff manifolds

TL;DR

This paper explores the topological properties of non-Hausdorff manifolds, focusing on homogeneous and everywhere non-Hausdorff cases, providing constructions, non-existence results, and detailed proofs using elementary topology and set theory.

Contribution

It introduces new constructions and non-existence results for various classes of non-Hausdorff manifolds, expanding understanding of their topological structure.

Findings

Homogeneous NH-manifolds can have non-homogeneous NH-sets.

Certain ENH-manifolds cannot be hereditarily separable under set theory.

No quasi-countably compact ENH-manifolds exist.

Abstract

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 $NH_X(x)$ 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 $NH_M(x)$ 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.