Page "Analytic set" Paragraph 14

The complement of an analytic set need not be analytic.

Suslin proved that if the complement of an analytic set is analytic then the set is Borel.

( Conversely any Borel set is analytic and Borel sets are closed under complements.

) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set containing one and disjoint from the other.

This is sometimes called the " Luzin separability principle " ( though it was implicit in the proof of Suslin's theorem ).