# Luzin separability principles

Two theorems in descriptive set theory, proved by N.N. Luzin in 1930 (see ). 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.