Descriptive Set Theory is the study of the properties and structure of definable sets of real numbers and, more generally, of definable subsets of \(\mathbb{R}^n\) and other Polish spaces (i.e., topological spaces that are homeomorphic to a separable complete metric space), such as the Baire space \(\mathcal{N}\) of all functions \(f:\mathbb{N} \to \mathbb{N}\), the space of complex numbers, Hilbert space, and separable Banach spaces. The simplest sets of real numbers are the basic open sets (i.e., the open intervals with rational endpoints), and their complements. The sets that are obtained in a countable number of steps by starting from the basic open sets and applying the operations of taking the complement and forming a countable union of previously obtained sets are the Borel sets. All Borel sets are regular, that is, they enjoy all the classical regularity properties. One example of a regularity property is the Lebesgue measurability: a set of reals is Lebesgue measurable if it differs from a Borel set by a null set, namely, a set that can be covered by sets of basic open intervals of arbitrarily-small total length. Thus, trivially, every Borel set is Lebesgue measurable, but sets more complicated than the Borel ones may not be. Other classical regularity properties are the Baire property (a set of reals has the Baire property if it differs from an open set by a meager set, namely, a set that is a countable union of sets that are not dense in any interval), and the perfect set property (a set of reals has the perfect set property if it is either countable or contains a perfect set, namely, a nonempty closed set with no isolated points). In ZFC one can prove that there exist non-regular sets of reals, but the AC is necessary for this (Solovay 1970).