In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.

Descriptive set theory begins with the study of Polish spaces and their Borel sets.

A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line ${\displaystyle \mathbb {R} }$, the Baire space ${\displaystyle {\mathcal {N}}}$, the Cantor space ${\displaystyle {\mathcal {C}}}$, and the Hilbert cube ${\displaystyle I^{\mathbb {N} }}$.

The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.

Because of these universality properties, and because the Baire space ${\displaystyle {\mathcal {N}}}$ has the convenient property that it is homeomorphic to ${\displaystyle {\mathcal {N}}^{\omega }}$, many results in descriptive set theory are proved in the context of Baire space alone.

The class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open sets of X. This means that the Borel sets of X are the smallest collection of sets such that:

A fundamental result shows that any two uncountable Polish spaces X and Y are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.

Each Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations of countable union and complementation must be used to obtain the set, beginning from open sets. The classification is in terms of countable ordinal numbers. For each nonzero countable ordinal α there are classes ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$, ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$, and ${\displaystyle \mathbf {\Delta } _{\alpha }^{0}}$.