  In this paper we show that {\omega}B- and {\omega}S-regular languages satisfy
the following separation-type theorem If L1,L2 are disjoint languages of
{\omega}-words both recognised by {\omega}B- (resp. {\omega}S)-automata then
there exists an {\omega}-regular language Lsep that contains L1, and whose
complement contains L2. In particular, if a language and its complement are
recognised by {\omega}B- (resp. {\omega}S)-automata then the language is
{\omega}-regular. The result is especially interesting because, as shown by
Boja\'nczyk and Colcombet, {\omega}B-regular languages are complements of
{\omega}S-regular languages. Therefore, the above theorem shows that these are
two mutually dual classes that both have the separation property. Usually (e.g.
in descriptive set theory or recursion theory) exactly one class from a pair C,
Cc has the separation property. The proof technique reduces the separation
property for {\omega}-word languages to profinite languages using Ramsey's
theorem and topological methods. After that reduction, the analysis of the
separation property in the profinite monoid is relatively simple. The whole
construction is technically not complicated, moreover it seems to be quite
extensible. The paper uses a framework for the analysis of B- and S-regular
languages in the context of the profinite monoid that was proposed by
Toru\'nczyk.

Formal Languages and Automata Theory