On Borel subsets of generalized Baire spaces

Abstract

We develop Descriptive Set Theory in Generalized Baire Spaces without assuming $\hat{I}{º}^{<\hat{I}º}=\hat{I}º$. We point out that without this assumption the basic topological concepts of these spaces have to be slightly modified in order to obtain a meaningful theory. This modification has no effect if $\hat{I}{º}^{<\hat{I}º}=\hat{I}º$. After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality $\hat{I}º$ in the space $\hat{I}{º}^{\hat{I}}º$ are $\hat{I}º$-Borel in our generalized sense. It turns out that this question depends, as is the case when $\hat{I}{º}^{<\hat{I}º}=\hat{I}º$, on stability theoretic properties (structure vs. non-structure) of the first order theory of the model.