On the scope of the Effros theorem
Volume 258 / 2022
Fundamenta Mathematicae 258 (2022), 211-223
MSC: Primary 54H11, 54H05; Secondary 22F05, 03E15, 03E45, 03E60.
DOI: 10.4064/fm100-12-2021
Published online: 11 March 2022
Abstract
All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group $G$ is Effros (that is, every continuous transitive action of $G$ on a non-meager space is micro-transitive). We complete the picture by obtaining the following results:
$\bullet $ under $\mathsf{AC}$, there exists a non-Effros group,
$\bullet $ under $\mathsf{AD} $, every group is Effros,
$\bullet $ under $\mathsf {V=L}$, there exists a coanalytic non-Effros group.
The above counterexamples will be graphs of discontinuous homomorphisms.
