  The space of Lascar strong types, on some sort and relative to a given first
order theory T, is in general not a compact Hausdorff space. This paper has at
least three aims. First to show that spaces of Lascar strong types and other
related spaces such as the Lascar group, have well-defined Borel cardinalities.
The second is to compute the Borel cardinalities of the known examples as well
as some new examples. The third is to explore notions of definable map,
embedding and isomorphism between these and related quotient objects. The
motivation for writing this paper is the recent discovery, via definable
groups, of new examples of non G-compact first order theories.
