Most "spaces" intrinsically attached to first order theories T are compact Hausdorff, but there are exceptions such as spaces of "Lascar strong types" and the Lascar group in suitable exotic first order theories. I will discuss the treatment of such "spaces" from the descriptive set theoretic point of view, as well as introduce a notion strictly in between continuous reduction and Borel reduction. (Joint work with K. Krupinski and S. Solecki, but closely connected to old work with Casanovas, Lascar, Ziegler, and recent work with Conversano).