## Manifolds: Hausdorffness versus homogeneity.(English)Zbl 1140.57014

In this work an $$n$$-manifold is a topological space that is locally homeomorphic to $$\mathbb R^n$$. 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

Zbl 0079.17101
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### References:

 [1] Nicolas Bourbaki, General topology. Chapters 1 – 4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. Nicolas Bourbaki, General topology. Chapters 5 – 10, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. [2] D. B. Fuks and V. A. Rokhlin, Beginner’s course in topology, Universitext, Springer-Verlag, Berlin, 1984. Geometric chapters; Translated from the Russian by A. Iacob; Springer Series in Soviet Mathematics. · Zbl 0562.54003 [3] David Gauld, Strong contractibility, Indian J. Math. 25 (1983), no. 1, 29 – 32. · Zbl 0567.54011 [4] André Haefliger and Georges Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, Enseignement Math. (2) 3 (1957), 107 – 125 (French). · Zbl 0079.17101 [5] Peter Nyikos, The theory of nonmetrizable manifolds, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 633 – 684. · Zbl 0583.54002 [6] Mary Ellen Rudin and Phillip Zenor, A perfectly normal nonmetrizable manifold, Houston J. Math. 2 (1976), no. 1, 129 – 134. · Zbl 0315.54028 [7] M. Spivak. Differential Geometry, Vol. 1. Publish or Perish, New York, 1970. · Zbl 0202.52001 [8] Z. Szentmiklóssy, \?-spaces and \?-spaces under Martin’s axiom, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 1139 – 1145. [9] D. van Dantzig. Ueber topologisch homogene Kontinua. Fund. Math. 15 (1930), 102-125. · JFM 56.1130.01 [10] James W. Vick, Homology theory, 2nd ed., Graduate Texts in Mathematics, vol. 145, Springer-Verlag, New York, 1994. An introduction to algebraic topology. · Zbl 0789.55004
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