Descriptive set theory

Summary

Descriptive set theory studies the definable subsets of Polish spaces such as the real line, the Cantor space and the Baire space. It sorts sets by how complicated their definitions are, in the Borel hierarchy and the projective hierarchy. Its classical results show that simply defined sets, such as Borel and analytic sets, behave well. For example, they are Lebesgue measurable, and they are either countable or contain a perfect set. For more complex sets these questions depend on axioms beyond ZFC, such as determinacy and large cardinals.