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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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