Descriptive Set Theory

Descriptive set theory studies the complexity of definable subsets of Polish spaces — complete separable metric spaces such as the real numbers and the Baire space. It classifies sets according to how many alternating quantifiers (over the natural numbers) are required to define them, yielding the Borel hierarchy, the projective hierarchy, and beyond.

The field arose from the study of pathological functions and the question of which sets of reals could be explicitly