Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
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Non-Borel sets without axiom of choiceIs every sigma-algebra the Borel algebra of a topology?
Parts of Set Theory immune to independence
Partitioning $\mathbb{R}$ into $\aleph_1$ Borel sets
Partitioning $\mathbb{R}$ into $\aleph_1$ Borel sets
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