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 Mat. Sb., 2013, Volume 204, Number 4, Pages 127–160 (Mi msb8114)

Topology of actions and homogeneous spaces

K. L. Kozlov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Topologization of a group of homeomorphisms and its action provide additional possibilities for studying the topological space, the group of homeomorphisms, and their interconnections. The subject of the paper is the use of the property of $d$-openness of an action (introduced by Ancel under the name of weak micro-transitivity) in the study of spaces with various forms of homogeneity. It is proved that a $d$-open action of a Čech-complete group is open. A characterization of Polish SLH spaces using $d$-openness is given, and it is established that any separable metrizable SLH space has an SLH completion that is a Polish space. Furthermore, the completion is realized in coordination with the completion of the acting group with respect to the two-sided uniformity. A sufficient condition is given for extension of a $d$-open action to the completion of the space with respect to the maximal equiuniformity with preservation of $d$-openness. A result of van Mill is generalized, namely, it is proved that any homogeneous CDH metrizable compactum is the only $G$-compactification of the space of rational numbers for the action of some Polish group.
Bibliography: 39 titles.

Keywords: $G$-space, $G$-extension, coset space, strong local homogeneity, countable dense homogeneity.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÐÍÏ 2.1.1.3704

DOI: https://doi.org/10.4213/sm8114

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English version:
Sbornik: Mathematics, 2013, 204:4, 588–620

Bibliographic databases:

UDC: 515.122.4+512.546
MSC: Primary 54H15; Secondary 22A05, 54D35, 54E35

Citation: K. L. Kozlov, “Topology of actions and homogeneous spaces”, Mat. Sb., 204:4 (2013), 127–160; Sb. Math., 204:4 (2013), 588–620

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8114
• https://doi.org/10.4213/sm8114
• http://mi.mathnet.ru/eng/msb/v204/i4/p127

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This publication is cited in the following articles:
1. K. L. Kozlov, “Spectral decompositions of spaces induced by spectral decompositions of acting groups”, Topology Appl., 160:11 (2013), 1188–1205
2. V. G. Pestov, “A topological transformation group without non-trivial equivariant compactifications”, Adv. Math., 311 (2017), 1–17
3. K. L. Kozlov, “$\mathbb R$-factorizable $G$-spaces”, Topology Appl., 227 (2017), 146–164
4. Whittington K., “The Sin Property in Homeomorphism Groups”, Topology Appl., 251 (2019), 94–106
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