Lusin's separation theorem
{{Short description|For 2 disjoint analytic subsets of Polish space, there is a Borel set containing only one}}
{{otheruses4|the separation theorem|the theorem on continuous functions|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 = ∅.{{harv|Kechris|1995|p=87}}. It is named after Nikolai Luzin, who proved it in 1927.{{harv|Lusin|1927}}.
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.
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
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References
- {{Citation
| last = Kechris
| first = Alexander
| authorlink = Alexander S. Kechris
| title = Classical descriptive set theory
| place = Berlin–Heidelberg–New York
| publisher = Springer-Verlag
| series = Graduate Texts in Mathematics
| volume = 156
| year = 1995
| pages = [https://archive.org/details/classicaldescrip0000kech/page/ xviii+402]
| doi = 10.1007/978-1-4612-4190-4
| isbn = 978-0-387-94374-9
| mr = 1321597
| zbl = 0819.04002
| url = https://archive.org/details/classicaldescrip0000kech/page/
}} ({{isbn|3-540-94374-9}} for the European edition)
- {{Citation
| last = Lusin
| first = Nicolas
| authorlink = Nikolai Luzin
| title = Sur les ensembles analytiques
| journal = Fundamenta Mathematicae
| volume = 10
| pages = 1–95
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm10/fm1011.pdf
| year = 1927
| language = French
| jfm = 53.0171.05
}}.
Category:Descriptive set theory
Category:Theorems in the foundations of mathematics
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