Descriptive Set Theory is the study of the “definable subsets” of Polish spaces such as the Reals (IR), the Baire space (ω^ω), or the Cantor space (2^ω).

The idea behind generalized descriptive set theory, as it is now conceived, is to generalize classical descriptive set theory by replacing ω with an uncountable cardinal κ in all definitions and relevant notions, with special emphasis placed on the study of the generalized Baire space (κ^κ, endowed with the natural topology) and its definable subsets.

If the initial goal was to identify a generalization of the real line from a topological and descriptive set theoretic standpoint, generalized descriptive set theory is today a thriving area of set theory in its own right. Significant connections with other fields of mathematical logic have been discovered, such as stability notions in model theory — a remarkable result being the descriptive set-theoretical analogue of Shelah’s Main Gap Theorem by Mangraviti and Motto Ros.