Perfect mappings in topological groups, cross-complementary subsets and quotients

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 701-709
  • ISSN: 0010-2628

Abstract

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The following general question is considered. Suppose that G is a topological group, and F , M are subspaces of G such that G = M F . 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 F M = 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 × 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 δ -closed mapping is introduced. This leads to new results on topological groups which are P -spaces.

How to cite

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Arhangel'skii, Aleksander V.. "Perfect mappings in topological groups, cross-complementary subsets and quotients." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 701-709. <http://eudml.org/doc/249190>.

@article{Arhangelskii2003,
abstract = {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.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological group; quotient group; locally compact subgroup; quotient mapping; perfect mapping; paracompact $p$-space; metrizable group; countable tightness; topological group; quotient group; locally compact subgroup; quotient mapping; perfect mapping; paracompact -space},
language = {eng},
number = {4},
pages = {701-709},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Perfect mappings in topological groups, cross-complementary subsets and quotients},
url = {http://eudml.org/doc/249190},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - Perfect mappings in topological groups, cross-complementary subsets and quotients
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 701
EP - 709
AB - 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.
LA - eng
KW - topological group; quotient group; locally compact subgroup; quotient mapping; perfect mapping; paracompact $p$-space; metrizable group; countable tightness; topological group; quotient group; locally compact subgroup; quotient mapping; perfect mapping; paracompact -space
UR - http://eudml.org/doc/249190
ER -

References

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  5. Bourbaki N., Elements de Mathématique, Premiere Partie, Livre 3, Ch. 3, 3-me ed., Hermann, Paris, 1949. 
  6. Engelking, General Topology, Warszawa, 1977. Zbl0684.54001
  7. Filippov V.V., On perfect images of paracompact p -spaces, Soviet Math. Dokl. 176 (1967), 533-536. (1967) MR0222853
  8. Graev M.I., Theory of topological groups, 1, Uspekhi Mat. Nauk 5 (1950), 3-56. (1950) MR0036245
  9. Henriksen M., Isbell J.R., Some properties of compactifications, Duke Math. J. 25 (1958), 83-106. (1958) Zbl0081.38604MR0096196
  10. Ivanovskij L.N., On a hypothesis of P.S. Alexandroff, Dokl. Akad. Nauk SSSR 123 (1958), 785-786. (1958) 
  11. Michael E., A quintuple quotient quest, General Topology Appl. 2 (1972), 91-138. (1972) Zbl0238.54009MR0309045
  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. (1989) MR1019483

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