# Luzin separability principles

Luzin separation principles

Two theorems in descriptive set theory, proved by N.N. Luzin in 1930 (see [1]). Two sets $E$ and $E_1$ without common points, lying in a Euclidean space, are called $B$-separable or Borel separable if there are two Borel sets $H$ and $H_1$ without common points containing $E$ and $E_1$, respectively. The first Luzin separation principle states that two disjoint analytic sets (cf. $\mathcal A$-set; Analytic set) are always $B$-separable. Since there are two disjoint co-analytic sets (cf. $C\mathcal A$-set) that are $B$-inseparable, the following definition makes sense: Two sets $E_1$ and $E_2$ without common points are separable by means of co-analytic sets if there are two disjoint sets $H_1$ and $H_2$ containing $E_1$ and $E_2$, respectively, each of which is a co-analytic set. Luzin's second separation principle asserts that if from two analytic sets one removes their common part, then the remaining parts are always separable by means of co-analytic sets.

#### References

 [1] N.N. [N.N. Luzin] Lusin, "Leçons sur les ensembles analytiques et leurs applications" , Gauthier-Villars (1930)

Both principles are still valid when Euclidean space is replaced by a Polish space. The first separation principle, which was already implicitly proved by M.Ya. Suslin (1917) while proving that a set $H$ is Borel if and only if $H$ and its complement are analytic, has numerous applications in analysis. Following C. Kuratowski, the second one is generally stated now as a reduction theorem for co-analytic sets (i.e. complements of analytic sets): If $C_1$ and $C_2$ are two co-analytic sets, then there exist two disjoint co-analytic sets $D_1\subset C_1$ and $D_2\subset C_2$ such that $D_1\cup D_2=C_1\cup C_2$. This is related to the use of countable ordinals in descriptive set theory and has some deep applications in analysis. For more details and references see Descriptive set theory.