View source for Luzin separability principles

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''Luzin separation principles''

Two theorems in [[Descriptive set theory|descriptive set theory]], proved by N.N. Luzin in 1930 (see [[#References|[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. [[A-set|$\mathcal A$-set]]; [[Analytic set|Analytic set]]) are always $B$-separable. Since there are two disjoint co-analytic sets (cf. [[CA-set|$C\mathcal A$-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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,   "Leçons sur les ensembles analytiques et leurs applications" , Gauthier-Villars  (1930)</TD></TR></table>



====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|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. [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Luzin,   "Sur les ensembles analytiques"  ''Fund. Math.'' , '''10'''  (1927)  pp. 1–92</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.J. Jech,   "Set theory" , Acad. Press  (1978)  pp. 523ff  (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Y.N. Moschovakis,   "Descriptive set theory" , North-Holland  (1980)</TD></TR></table>