# Luzin sieve

An arbitrary mapping $W : \mathbf Q _ {0} \rightarrow 2 ^ {X}$ that puts each dyadic fraction $r \in \mathbf Q _ {0}$ into correspondence with a subset $W _ {r} \subset X$. As a rule, $X$ is assumed to be a complete separable metric space. It was introduced by N.N. Luzin [1]. The set $A$ of points $x \in X$ such that there is an infinite sequence $r _ {1} < r _ {2} < \dots$ that satisfies the condition $x \in W _ {r _ {1} } \cap W _ {r _ {2} } \cap \dots$ is said to be sifted through the Luzin sieve $W$. For every ${\mathcal A}$- operation there is a Luzin sieve $W$ such that the result of this ${\mathcal A}$- operation is sifted through $W$. The main result concerning the Luzin sieve is that a Luzin set of the $n$- th class (or of the projective class $L _ {n}$) is invariant under the operation of sifting through the Luzin sieve for $0 \neq n \neq 2$.

#### References

 [1] N.N. Luzin, "Sur les ensembles analytiques" Fund. Math. , 10 (1927) pp. 1–95 [2] C. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)

If $A$ is an analytic set and $W = \{ {W _ {r} } : {r \in \mathbf Q } \}$ is a Luzin sieve for $A$ consisting of closed sets, then, as one readily sees, $X \setminus A = \{ {x } : {M _ {x} \textrm{ is well\AAh ordered by } \geq } \}$, where $M _ {x} = \{ {r } : {x \in W _ {r} } \}$. The sets $A _ \alpha = \{ {x \in X \setminus A } : {\textrm{ the order type of } M _ {x } \textrm{ is } \alpha } \}$, where $\alpha < \omega _ {1}$, are called the constituents of the set $X \setminus A$ determined by the sieve $W$.