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 are continuous images of Luzin sets of class
. The Luzin sets of class
are complements of Luzin sets of class
. In particular, Luzin sets of class 1, that is, continuous images of Borel sets, are called analytic sets, or
-sets or Suslin sets (cf.
-set; Analytic set). The concept of a Luzin set is due to N.N. Luzin [1]. If the sets
are Luzin sets of class
, then
and
are also Luzin sets of class
. If the sets
are Luzin sets of class
lying in complete separable metric spaces
, then the direct product (finite or countable)
is a Luzin set of class
in the space
. A Luzin set of odd class
situated in a space
coincides with the projection of a set of class
situated in
. The space
of irrational numbers in the interval
contains, for any
, a Luzin set of class
that is not a Luzin set of class
; the space
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 are generally called
-sets and those of class
are called
-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) |
Luzin set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_set&oldid=47723