Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{\omega_1\omega}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{\omega_1\omega… Expand

Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets… Expand

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the… Expand

the descriptive set theory of polish group actions are a good way to achieve details about operating certainproducts. Many products that you buy can be obtained using instruction manuals. These user… Expand