Abstract
For each countable ordinal $\xi$ and pair $({A_0}, {A_1})$ of disjoint analytic subsets of ${2^\omega }$, we define a closed game ${J_\xi }({A_0}, {A_1})$ and a complete $\Pi _\xi ^0$ subset ${H_\xi }$ of ${2^\omega }$ such that (i) a winning strategy for player I constructs a $\sum _\xi ^0$ set separating ${A_0}$ from ${A_1}$; and (ii) a winning strategy for player II constructs a continuous map $\varphi :{2^\omega } \to {A_0} \cup {A_1}$ with ${\varphi ^{ - 1}}({A_0}) = {H_\xi }$. Applications of this construction include: A proof in second order arithmetics of the statement "every $\Pi _\xi ^0$ non $\sum _\xi ^0$ set is $\Pi _\xi ^0$-complete"; an extension to all levels of a theorem of Hurewicz about $\sum _2^0$ sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieffâs and Wadgeâs hierarchies.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.