Suppose $G$ is a compact Hausdorff (not necessarily Polish) group, acting continuously on a Polish (not necessarily compact) space $X$ (i.e. the map $G\times X\to X$ is continuous).

Is it true that there is a compact Polish group $H$ acting on $X$ through which the action of $G$ factors?

This is seems true if $X$ is compact, because the action induces a continuous homomorphism from $G$ into the homeomorphism group of $X$, which is Polish in this case, so we can take the image of $G$ for $H$. I suppose more general conditions which force the homeomorphism group to be a Polish group with compact-open topology would suffice as well, but I have no idea about the general case.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.