Lusin's separation theorem
For 2 disjoint analytic subsets of Polish space, there is a Borel set containing only one
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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 (An) of disjoint analytic sets there is a sequence (Bn) of disjoint Borel sets such that An ⊆ Bn 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.
