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 = ∅.^[1] It is named after Nikolai Luzin, who proved it in 1927.^[2]

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. ^[1]

An immediate consequence is Suslin's theorem, which states that if a set and its complement are both analytic, then the set is Borel.