descriptive set theory 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 oth ...

and

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, Lusin's separation theorem states that if ''A'' and ''B'' are disjoint analytic subsets of

Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named ...

, then there is a

Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are name ...

''C'' in the space such that ''A'' ⊆ ''C'' and ''B'' ∩ ''C'' = ∅.. It is named after

Nikolai Luzin Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/R ...

, who proved it in 1927.. The theorem can be generalized to show that for each sequence (''A''_''n'') of disjoint analytic sets there is a sequence (''B''_''n'') of disjoint Borel sets such that ''A''_''n'' ⊆ ''B''_''n'' for each ''n''. An immediate consequence is Suslin's theorem, which states that if a set and its complement are both analytic, then the set is Borel.

Notes

References

* ( for the European edition) *. Descriptive set theory Theorems in the foundations of mathematics Theorems in topology {{mathlogic-stub