Luzin set
Encyclopedia
In real analysis
and descriptive set theory
, a Luzin set (or Lusin set), named for N. N. Luzin, is an uncountable subset A of the reals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is an uncountable set of reals which meets every first category set in only countably many points. Luzin
proved that, if the continuum 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.
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
and descriptive set theory
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
, a Luzin set (or Lusin set), named for N. N. Luzin, is an uncountable subset A of the reals such that every uncountable subset of A is nonmeager; that is, of second Baire category. Equivalently, A is an uncountable set of reals which meets every first category set in only countably many points. Luzin
Nikolai Luzin
Nikolai Nikolaevich Luzin, , was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the...
proved that, if the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
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
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
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.