# Article

 Title: Two types of remainders of topological groups (English) Author: Arhangel'skii, A. V. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 49 Issue: 1 Year: 2008 Pages: 119-126 . Category: math . Summary: We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel'skii, {\it Some connections between properties of topological groups and of their remainders\/}, Moscow Univ. Math. Bull. 54:3 (1999), 1--6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions. (English) Keyword: remainder Keyword: compactification Keyword: topological group Keyword: $p$-space Keyword: Lindelöf $p$-space Keyword: metrizability Keyword: countable type Keyword: Lindelöf space Keyword: pseudocompact space Keyword: $\pi$-base Keyword: compactification MSC: 54A25 MSC: 54B05 idZBL: Zbl 1212.54086 idMR: MR2433629 . Date available: 2009-05-05T17:06:53Z Last updated: 2013-09-22 Stable URL: http://hdl.handle.net/10338.dmlcz/119706 . 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.: Some connections between properties of topological groups and of their remainders.Moscow Univ. Math. Bull. 54:3 (1999), 1-6. MR 1711899 Reference: [4] Arhangel'skii A.V.: Topological invariants in algebraic environment.in: Recent Progress in General Topology 2, eds. M. Hušek, Jan van Mill, North-Holland, Amsterdam, 2002, pp.1-57. Zbl 1030.54026, MR 1969992 Reference: [5] 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: [6] Arhangel'skii A.V.: More on remainders close to metrizable spaces.Topology Appl. 154 (2007), 1084-1088. Zbl 1144.54001, MR 2298623, 10.1016/j.topol.2006.10.008 Reference: [7] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 Reference: [8] Filippov V.V.: On perfect images of paracompact $p$-spaces.Soviet Math. Dokl. 176 (1967), 533-536. MR 0222853 Reference: [9] Henriksen M., Isbell J.R.: Some properties of compactifications.Duke Math. J. 25 (1958), 83-106. Zbl 0081.38604, MR 0096196, 10.1215/S0012-7094-58-02509-2 Reference: [10] Tkachenko M.G.: The Suslin property in free topological groups over compact spaces (Russian).Mat. Zametki 34 (1983), 601-607; English translation: Math. Notes 34 (1983), 790-793. MR 0722229 Reference: [11] Roelke W., Dierolf S.: Uniform Structures on Topological Groups and their Quotients.McGraw-Hill, New York, 1981. .

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