Lusin's separation 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.