Lusin's separation theorem
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Short description: For 2 disjoint analytic subsets of Polish space, there is a Borel set containing only one
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.
Notes
References
- Kechris, Alexander (1995), Classical descriptive set theory, Graduate Texts in Mathematics, 156, Berlin–Heidelberg–New York: Springer-Verlag, pp. xviii+402, doi:10.1007/978-1-4612-4190-4, ISBN 978-0-387-94374-9, https://archive.org/details/classicaldescrip0000kech/page/ (ISBN 3-540-94374-9 for the European edition)
- Lusin, Nicolas (1927), "Sur les ensembles analytiques" (in French), Fundamenta Mathematicae 10: 1–95, http://matwbn.icm.edu.pl/ksiazki/fm/fm10/fm1011.pdf.
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