Topology of actions and homogeneous spaces (Q2841130)

The subject of this interesting work is the use of the property of \(d\)-openness of an action in the study of spaces with various forms of homogeneity. The concept of \(d\)-openness was introduced by \textit{F. D. Ancel} in [Mich. Math. J. 34, 39--55 (1987; Zbl 0626.54036)] under the name of weak microtransitivity. The author proves that a \(d\)-open action of a Čech-complete group is open. He gives a characterization of Polish strongly locally homogeneous spaces (SLH) using \(d\)-openness and he shows 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. The author gives a sufficient condition for extension of a \(d\)-open action to the completion of the space with respect to the maximal equiuniformity with preservation of \(d\)-openness. He also generalizes a result of \textit{J. van Mill} [Monatsh. Math. 157, No. 3, 257--266 (2009; Zbl 1172.54025)], namely, he proves that any countable dense homogeneous metrizable compactum is the only \(G\)-compactification of the space of rational numbers for the action of some Polish group.