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)

Comments

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.

Under extra set-theoretical hypotheses (Gödel's constructibility axiom, large-cardinal and, especially, determinacy hypotheses), much more is known nowadays on the separation principle at higher levels of the projective hierarchy, cf. [a3], [a4].

References

[a1]	N.N. Luzin, "Sur les ensembles analytiques" Fund. Math. , 10 (1927) pp. 1–92
[a2]	K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
[a3]	T.J. Jech, "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)
[a4]	Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)