  Descriptive set theory is mainly concerned with studying subsets of the space
of all countable binary sequences. In this paper we study the generalization
where countable is replaced by uncountable. We explore properties of
generalized Baire and Cantor spaces, equivalence relations and their Borel
reducibility. The study shows that the descriptive set theory looks very
different in this generalized setting compared to the classical, countable
case. We also draw the connection between the stability theoretic complexity of
first-order theories and the descriptive set theoretic complexity of their
isomorphism relations. Our results suggest that Borel reducibility on
uncountable structures is a model theoretically natural way to compare the
complexity of isomorphism relations.
