Lusin's separation theorem

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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 (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.


  1. ^ a b (Kechris 1995, p. 87).
  2. ^ (Lusin 1927).