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Abstract

We investigate various aspects of non-Hausdorff manifolds (NH-manifold for short). First, we extend some known results about which covering properties, together with (a weakening of) homogeneity, imply that a manifold is Hausdorff. Let $NH_M(x)$ be the subset of points of a space $M$ which cannot be separated of $x$ by open sets. We give various properties that imply discreteness of $NH_M(x)$. Using minimal flows, we exhibit homogeneous NH-manifolds $M$ with non-homogeneous (and thus non-discrete) $NH_M(x)$. In particular, $NH_M(x)$ may be a coutable discrete union of lines and compact intervals (in dimension $2$) or $n$-dimensional torus (in dimension $n+1$). We show that there is an everywhere non-Hausdorff hereditarily separable manifold under CH and that the existence of an everywhere non-Hausdorff hereditarily separable manifold in a particular class implies that of a locally compact S-space. We also construct various examples of somewhat pathological NH-manifolds. In particular, we show that there are NH-manifolds $M$ with a point $x\in M$ such that $NH_M(x)$ is a copy of: the Cantor space, any special tree, any $\Psi$-space (dimension $1$), $\mathbb{R}$, the long ray, (part of) boundaries of domains in $\mathbb{R}^2$ (dimension $2$). We also build a homogeneous $1$-dimensional NH-manifold $M$ such that for each $x,y\in M$ there is some $z$ with $x\in NH_M(z)$, $z\in NH_M(y)$. We start by recalling basic properties of NH-manifolds and 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) for some of the examples. 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.