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Zbl 1140.57014
Baillif, Mathieu; Gabard, Alexandre
Manifolds: Hausdorffness versus homogeneity. (English)
[J] Proc. Am. Math. Soc. 136, No. 3, 1105-1111 (2008). ISSN 0002-9939; ISSN 1088-6826/e

In this work an $n$-manifold is a topological space that is locally homeomorphic to $\Bbb R^n$. 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. \par The first example, called the ``complete feather,'' was first defined by {\it A. Haefliger} and {\it 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.
[Leonhard R. Rubin (Norman)]

MSC 2000:: *57N99 Topological manifolds
54D10 Lower separation axioms
54E52 Baire category, Baire spaces

Keywords: Baire space; contractible; homogeneous; Lindel\" of; manifold; non-Hausdorff; separable

Citations: Zbl 0079.17101

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