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.
@article{119425,
author = {Aleksander V. Arhangel'skii},
title = {Perfect mappings in topological groups, cross-complementary subsets and quotients},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {701-709},
zbl = {1098.22003},
mrnumber = {2062887},
language = {en},
url = {http://dml.mathdoc.fr/item/119425}
}
Arhangel'skii, Aleksander V. Perfect mappings in topological groups, cross-complementary subsets and quotients. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 701-709. http://gdmltest.u-ga.fr/item/119425/
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. (1965) | MR 0190889
Quotients with respect to locally compact subgroups, to appear in Houston J. Math. | MR 2123011 | Zbl 1077.54022
Bisequential spaces, tightness of products, and metrizability conditions in topological groups, Trans. Moscow Math. Soc. 55 (1994), 207-219. (1994) | MR 1468459
Fundamentals of General Topology: Problems and Exercises, Reidel, 1984. | MR 0785749
Elements de Mathématique, Premiere Partie, Livre 3, Ch. 3, 3-me ed., Hermann, Paris, 1949.
General Topology, Warszawa, 1977. | Zbl 0684.54001
On perfect images of paracompact $p$-spaces, Soviet Math. Dokl. 176 (1967), 533-536. (1967) | MR 0222853
Theory of topological groups, 1, Uspekhi Mat. Nauk 5 (1950), 3-56. (1950) | MR 0036245
Some properties of compactifications, Duke Math. J. 25 (1958), 83-106. (1958) | MR 0096196 | Zbl 0081.38604
On a hypothesis of P.S. Alexandroff, Dokl. Akad. Nauk SSSR 123 (1958), 785-786. (1958)
A quintuple quotient quest, General Topology Appl. 2 (1972), 91-138. (1972) | MR 0309045 | Zbl 0238.54009
Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill, New York, 1981.
Topological groups and Dugundji spaces, Mat. Sb. 180:8 (1989), 1092-1118. (1989) | MR 1019483