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Manifolds: Hausdorffness versus homogeneity. (English) Zbl 1140.57014

In this work an $n$-manifold is a topological space that is locally homeomorphic to ${ℝ}^{n}$. The objective is to analyze the relationship between homogeneity and being Hausdorff among such spaces. Is homogeneity a sufficient condition to characterize those manifolds that are Hausdorff? The authors exhibit two examples which show that the answer is no.

The first example, called the “complete feather,” was first defined by A. Haefliger and G. Reeb [Enseign. Math., II. Sér. 3, 107–125 (1957; Zbl 0079.17101)]. This space is a connected non-Hausdorff homogeneous 1-manifold which is neither separable nor Lindelöf. It is contractible but it does not admit a strong deformation retraction to any of its points. The second example is called the “everywhere doubled line.” It is a connected, homogeneous and separable 1-manifold that is neither Hausdorff nor Lindelöf.

##### MSC:
 57N99 Topological manifolds 54D10 Lower separation axioms (${T}_{0}$–${T}_{3}$, etc.) 54E52 Baire category, Baire spaces