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

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 (κκ, 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.