# Article

 Title: The Baire property in remainders of topological groups and other results  (English) Author: Arhangel'skii, Alexander Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 50 Issue: 2 Year: 2009 Pages: 273-279 Summary lang: English . Category: math . Summary: It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces $G$ and $H$ are homeomorphic. Some other corollaries and related results are presented. Keyword: Baire property Keyword: $\sigma$-compact Keyword: Čech-complete space Keyword: compactification Keyword: Čech-Stone compactification Keyword: Rajkov complete Keyword: paracompact $p$-space MSC: 54B05 MSC: 54H11 MSC: 54H15 idZBL: Zbl 1212.54098 idMR: MR2537836 . Date available: 2009-08-18T12:25:08Z Last updated: 2013-09-22 Stable URL: http://hdl.handle.net/10338.dmlcz/133433 . Reference: [1] Arhangel'skii A.V.: On a class of spaces containing all metric and all locally compact spaces.Mat. Sb. 67 (109) (1965), 55--88; English translation: Amer. Math. Soc. Transl. 92 (1970), 1--39. MR 0190889 Reference: [2] Arhangel'skii A.V.: Classes of topological groups.Russian Math. Surveys 36 (3) (1981), 151-174. MR 0622722, 10.1070/RM1981v036n03ABEH004249 Reference: [3] Arhangel'skii A.V.: Moscow spaces and topological groups.Topology Proc. 25 (2000), 383--416. Zbl 1027.54038, MR 1925695 Reference: [4] Arhangel'skii A.V.: Remainders in compactifications and generalized metrizability properties.Topology Appl. 150 (2005), 79--90. Zbl 1075.54012, MR 2133669, 10.1016/j.topol.2004.10.015 Reference: [5] Arhangel'skii A.V.: Two types of remainders of topological groups.Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119--126. MR 2433629 Reference: [6] Arhangel'skii A.V., Ponomarev V.I.: Fundamentals of General Topology in Problems and Exercises.Reidel, 1984 (translated from Russian). MR 0785749 Reference: [7] Arhangel'skii A.V., Tkachenko M.G.: Topological Groups and Related Structures.Atlantis Press, Paris; World Scientific, Hackensack, NJ, 2008. MR 2433295 Reference: [8] Choban M.M.: On completions of topological groups.Vestnik Moskov. Univ. Ser. Mat. Mech. 1 (1970), 33--38 (in Russian). MR 0279226 Reference: [9] Choban M.M.: Topological structure of subsets of topological groups and their quotients.in Topological Structures and Algebraic Systems, Shtiintsa, Kishinev, 1977, pp. 117--163 (in Russian). Reference: [10] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 Reference: [11] Rančin D.V.: Tightness, sequentiality, and closed covers.Dokl. Akad. Nauk SSSR 232 (1977), 1015--1018. MR 0436074 .

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