# Article

Full entry | PDF   (0.2 MB)
Keywords:
topological group; quotient group; locally compact subgroup; quotient mapping; perfect mapping; paracompact $p$-space; metrizable group; countable tightness
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.
References:
[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
[2] Arhangel'skii A.V.: Quotients with respect to locally compact subgroups. to appear in Houston J. Math. MR 2123011 | Zbl 1077.54022
[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
[4] Arhangel'skii A.V., Ponomarev V.I.: Fundamentals of General Topology: Problems and Exercises. Reidel, 1984. MR 0785749
[5] Bourbaki N.: Elements de Mathématique, Premiere Partie, Livre 3, Ch. 3. 3-me ed., Hermann, Paris, 1949.
[6] Engelking: General Topology. Warszawa, 1977. Zbl 0684.54001
[7] Filippov V.V.: On perfect images of paracompact $p$-spaces. Soviet Math. Dokl. 176 (1967), 533-536. MR 0222853
[8] Graev M.I.: Theory of topological groups, 1. Uspekhi Mat. Nauk 5 (1950), 3-56. MR 0036245
[9] Henriksen M., Isbell J.R.: Some properties of compactifications. Duke Math. J. 25 (1958), 83-106. MR 0096196 | Zbl 0081.38604
[10] Ivanovskij L.N.: On a hypothesis of P.S. Alexandroff. Dokl. Akad. Nauk SSSR 123 (1958), 785-786.
[11] Michael E.: A quintuple quotient quest. General Topology Appl. 2 (1972), 91-138. MR 0309045 | Zbl 0238.54009
[12] Roelke W., Dierolf S.: Uniform Structures on Topological Groups and Their Quotients. McGraw-Hill, New York, 1981.
[13] Uspenskij V.V.: Topological groups and Dugundji spaces. Mat. Sb. 180:8 (1989), 1092-1118. MR 1019483

Partner of