Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

Streszczenie

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.