descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), 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 a ...

and

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, 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 space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...

, then there is a

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

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

Nikolai Luzin Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Sov ...

, 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