Lusin's separation theorem

This article is about the separation theorem. For the theorem on continuous functions, see Lusin's theorem.

In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅. It is named after Nikolai Luzin, 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.

References

• Kechris, Alexander (1995), Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Berlin–Heidelberg–New York: Springer-Verlag, pp. xviii+402, doi:10.1007/978-1-4612-4190-4, ISBN 978-0-387-94374-9, MR 1321597, Zbl 0819.04002 (ISBN 3-540-94374-9 for the European edition)
• Lusin, Nicolas (1927), "Sur les ensembles analytiques" (PDF), Fundamenta Mathematicae (in French), 10: 1–95, JFM 53.0171.05.