# Article

 Title: Perfect mappings in topological groups, cross-complementary subsets and quotients  (English) Author: Arhangel'skii, A. V. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 44 Issue: 4 Year: 2003 Pages: 701-709 . Category: math . Summary: The following general question is considered. Suppose that $G$ is a topological group, and $F$, $M$ are subspaces of $G$ such that $G=MF$. Under these general assumptions, how are the properties of $F$ and $M$ related to the properties of $G$? For example, it is observed that if $M$ is closed metrizable and $F$ is compact, then $G$ is a paracompact $p$-space. Furthermore, if $M$ is closed and first countable, $F$ is a first countable compactum, and $FM=G$, then $G$ is also metrizable. Several other results of this kind are obtained. An extensive use is made of the following old theorem of N. Bourbaki [5]: if $F$ is a compact subset of a topological group $G$, then the natural mapping of the product space $G\times F$ onto $G$, given by the product operation in $G$, is perfect (that is, closed continuous and the fibers are compact). This fact provides a basis for applications of the theory of perfect mappings to topological groups. Bourbaki's result is also generalized to the case of Lindelöf subspaces of topological groups; with this purpose the notion of a $G_\delta$-closed mapping is introduced. This leads to new results on topological groups which are $P$-spaces. Keyword: topological group Keyword: quotient group Keyword: locally compact subgroup Keyword: quotient mapping Keyword: perfect mapping Keyword: paracompact $p$-space Keyword: metrizable group Keyword: countable tightness MSC: 22A05 MSC: 54A05 MSC: 54D35 MSC: 54D60 MSC: 54H11 idZBL: Zbl 1098.22003 idMR: MR2062887 . Date available: 2009-01-08T19:32:23Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119425 . Reference: [1] Arhangelskii A.V.: On a class of spaces containing all metric and all locally bicompact spaces.Mat. Sb. (N.S.) 67 (109) (1965), 55-88; English translation: Amer. Math. Soc. Transl. 92 (1970), 1-39. MR 0190889 Reference: [2] Arhangel'skii A.V.: Quotients with respect to locally compact subgroups.to appear in Houston J. Math. Zbl 1077.54022, MR 2123011 Reference: [3] Arhangel'skii A.V.: Bisequential spaces, tightness of products, and metrizability conditions in topological groups.Trans. Moscow Math. Soc. 55 (1994), 207-219. MR 1468459 Reference: [4] Arhangel'skii A.V., Ponomarev V.I.: Fundamentals of General Topology: Problems and Exercises.Reidel, 1984. MR 0785749 Reference: [5] Bourbaki N.: Elements de Mathématique, Premiere Partie, Livre 3, Ch. 3.3-me ed., Hermann, Paris, 1949. Reference: [6] Engelking: General Topology.Warszawa, 1977. Zbl 0684.54001 Reference: [7] Filippov V.V.: On perfect images of paracompact $p$-spaces.Soviet Math. Dokl. 176 (1967), 533-536. MR 0222853 Reference: [8] Graev M.I.: Theory of topological groups, 1.Uspekhi Mat. Nauk 5 (1950), 3-56. MR 0036245 Reference: [9] Henriksen M., Isbell J.R.: Some properties of compactifications.Duke Math. J. 25 (1958), 83-106. Zbl 0081.38604, MR 0096196 Reference: [10] Ivanovskij L.N.: On a hypothesis of P.S. Alexandroff.Dokl. Akad. Nauk SSSR 123 (1958), 785-786. Reference: [11] Michael E.: A quintuple quotient quest.General Topology Appl. 2 (1972), 91-138. Zbl 0238.54009, MR 0309045 Reference: [12] Roelke W., Dierolf S.: Uniform Structures on Topological Groups and Their Quotients.McGraw-Hill, New York, 1981. Reference: [13] Uspenskij V.V.: Topological groups and Dugundji spaces.Mat. Sb. 180:8 (1989), 1092-1118. MR 1019483 .

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