Descriptive set theory

In mathematical logic, descriptive set theory 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. Equivalently, 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 ${\displaystyle I^{\mathbb {N} }}$.

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