# Luzin set

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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)
How to Cite This Entry:
Luzin set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_set&oldid=47723
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article