On non-Hausdorff manifolds
Mathieu Baillif
TL;DR
This work systematically analyzes non-Hausdorff manifolds (NH-manifolds), focusing on homogeneous or near-homogeneous cases (HNH) and the stronger everywhere non-Hausdorff condition (ENH). It develops a framework around the invariant $NH_X(x)$, establishes when NH-structures must be discrete, and studies how strengthened covering properties, sorted neighborhoods, rim-simplicity, and specific homeomorphisms constrain non-Hausdorff behaviour. The paper constructionally explores ENH-manifolds, notably towel-rack manifolds, and demonstrates the existence of hereditarily separable ENH-manifolds under CH via Kunen-line techniques, linking to S-spaces and set-theoretic hypotheses. It also investigates quasi-countable compactness, providing both negative results for ENH-manifolds and intricate counterexamples (including Nyikos-style sprats) that refine our understanding of NH phenomena in dimension 1 and 2. Collectively, the results illustrate the rich and sometimes pathological landscape of NH-manifolds, showing how homogeneity, combinatorics, and set theory shape their global structure and the possible forms of $NH_M(x)$.
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 (sometimes 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.