# Luzin 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 . 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.