I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical
examples are R^n, C^n, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2^N, the Baire space N^N, the infinite symmetric group S_∞, the unitary group (of the Hilbert space), the group of
measure preserving transformations of the unit interval, etc
