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.