Luzin set

projective set

A subset of a complete separable metric space, defined by induction as follows. The Luzin sets of class 0 are the Borel sets (cf. Borel set). The Luzin sets of class $ 2 n + 1 $ are continuous images of Luzin sets of class $ 2 n $. The Luzin sets of class $ 2 n $ are complements of Luzin sets of class $ 2 n - 1 $. In particular, Luzin sets of class 1, that is, continuous images of Borel sets, are called analytic sets, or $ {\mathcal A} $- sets or Suslin sets (cf. $ {\mathcal A} $- set; Analytic set). The concept of a Luzin set is due to N.N. Luzin [1]. If the sets $ P _ {i} $ are Luzin sets of class $ n $, then $ \cup _ {i=} 1 ^ {k} P _ {i} $ and $ \cap _ {i=} 1 ^ {k} P _ {i} $ are also Luzin sets of class $ n $. If the sets $ P _ {i} \subset X _ {i} $ are Luzin sets of class $ n $ lying in complete separable metric spaces $ X _ {i} $, then the direct product (finite or countable) $ \prod _ {i} P _ {i} $ is a Luzin set of class $ n $ in the space $ \prod _ {i} X _ {i} $. A Luzin set of odd class $ n $ situated in a space $ X $ coincides with the projection of a set of class $ n- 1 $ situated in $ X \times X $. The space $ X $ of irrational numbers in the interval $ [ 0 , 1 ] $ contains, for any $ n> 0 $, a Luzin set of class $ n $ that is not a Luzin set of class $ < n $; the space $ X $ also contains sets that are not Luzin sets.

References

[1]	N.N. [N.N. Luzin] Lusin, "Sur un problème de M. Emile Borel et les ensemble projectifs de M. Henri Lebesgue" C.R. Acad. Sci. Paris , 180 (1925) pp. 1318–1320
[2]	N.N. [N.N. Luzin] Lusin, "Leçons sur les ensembles analytiques et leurs applications" , Gauthier-Villars (1930)
[3]	K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)

Comments

The term "Luzin set" is in the West almost exclusively used for a subset of the real line whose intersection with every nowhere-dense set is countable (see Luzin space). The sets discussed in the main article above are almost exclusively called projective sets (cf. Projective set). The sets of class $ 2 n + 1 $ are generally called $ \Sigma _ {n} ^ {1} $- sets and those of class $ 2 n $ are called $ \Pi _ {n} ^ {1} $- sets. See Descriptive set theory.

All important problems about projective sets have received satisfactory answers during the last three decades, see Descriptive set theory and Luzin problem.

References

[a1]	Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)
[a2]	T.J. Jech, "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)