- Luzin set
In
real analysis anddescriptive set theory , a Luzin set (or Lusin set), named forN. N. Luzin , is anuncountable subset A of the reals such that every uncountable subset of A isnonmeager ; that is, of secondBaire category . Equivalently, A is an uncountable set of reals which meets every first category set in only countably many points. Luzin proved that, if thecontinuum hypothesis holds, then every nonmeager set has a Luzin subset.A Luzin space (or Lusin space) is an uncountable topological T1-space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a countable or even arbitrary number of isolated points. (Warning: the term "Lusin space" also has an unrelated meaning in general topology as the image of a separable complete metric space under a continuous map.)Assuming
Martin's Axiom and the negation of the Continuum Hypothesis, there are no Luzin spaces (or Luzin sets).Obvious properties of a Luzin set are that it must be nonmeager (otherwise the set itself is an uncountable meager subset) and of
measure zero , because every set of positive measure contains a meager set which also has positive measure, and is therefore uncountable.The
measure-category duality provides a measure analogue of Luzin sets - sets of positive measure, every uncountable subset of which has positive outer measure.References
* citation|doi= 10.1070/RM1978v033n06ABEH003884
first=A V |last=Arkhangelskii|title=STRUCTURE AND CLASSIFICATION OF TOPOLOGICAL SPACES AND CARDINAL INVARIANTS|journal= RUSS MATH SURV|year= 1978|volume= 33 |issue=6|pages= 33-96|url=http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=3884 Paper mentioning Luzin spaces
*springer|title=Luzin space|id=l/l061110|first=B.A.|last= Efimov
*citation|first=N.N. |last= Lusin|title=Sur un problème de M. Baire|journal= C.R. Acad. Sci. Paris |volume= 158 |year=1914|pages= 1258–1261
*citation|first=John C.|last= Oxtoby |title=Measure and category: a survey of the analogies between topological and measure spaces |publisher=Springer-Verlag |location=Berlin |year=1980 |pages= |isbn=0-387-90508-1 |oclc= |doi=
Wikimedia Foundation. 2010.
