Orbit cardinals: On the effective cardinalities arising as quotient spaces of the form \(X/G\) where \(G\) acts on a Polish space \(X\)
The author considers the effective cardinalities that arise from the continuous actions of Polish groups. The first result is a generalization of Glimm-Effros dichotomy for Polish group actions. Assuming AD\(^{L({\mathbb R})}\), if \(G\) is a Polish group acting continuously on a Polish space \(X\) and \(A\subset X\) is in \(L({\mathbb R})\), then either \(|A/G|_{L({\mathbb R})}\leq |2^{<\omega_1}|_{L({\mathbb R})}\), or \(|{\mathbb R}/{\mathbb Q}|_{L({\mathbb R})}< |A/G|_{L({\mathbb R})}\) where \(A/G\) is the set of all orbits intersecting \(A-\{g\cdot a:g\in G\), \(a\in A\}\). Another direction was suggested by H. Becker, and the author proves in ZFC that if \(G\) is a Polish group with a left invariant complete metric acting continuously on a Polish space \(X\) and \(A\subset X\) is \({\underset\sim\Sigma}^1_1\), then either there is a \({\underset\sim\Delta}^1_2\) function \(\theta:A\to 2^\omega\) such that for all \(x_1,x_2\in A\), \(\exists g\in G\) \((g\cdot x_1=g\cdot x_2\Leftrightarrow \theta(x_1)=\theta(x_2))\), or there is a Borel \(\theta:{\mathbb R}\to A\) such that for all \(r_1,r_2\in\mathbb R\), \(r_1-r_2\in{\mathbb Q}\Leftrightarrow\exists g\in G\) \((g\cdot\theta(r_1)=\theta(r_2))\). The proof of this theorem yields, under AD\(^{L({\mathbb R})}\), a generalization which enables us to replace \(2^{<\omega_1}\) by \(2^\omega\) in the first mentioned result. By the same method one obtains that the orbit structure of a complete left invariant metric group never reduces the equality relation on countable sets of reals. Finally, answering a question whether Polish groups admitting left invariant complete metric are the only Polish groups satisfying Vaught's conjecture the author proves that there is a Polish group \(G\) with no compatible left invariant metric such that whenever it acts continuously on a Polish space \(X\), then either \(|X/G|\leq\aleph_0\) or \(2^{\aleph_0}\leq|X/G|\).
generalization of Glimm-Effros dichotomy
scientific article; zbMATH DE number 1356478
scientific article