Topology of actions and homogeneous spaces

K. L. Kozlov

Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 588-620

Résumé

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.

Détail

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