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 = ∅ (Kechris 1995, p. 87). It is named after Nicolas Lusin, 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⊆A_n for each n (Kechris 1995, p. 87).

References

• Alexander Kechris (1995). Classical descriptive set theory. Graduate texts in mathematics. Vol. 156. ISBN 0387943749.
• Nicolas Lusin (1927). "Sur les ensembles analytiques" (PDF). Fundamenta Mathematicae (in French). 10: 1–95.