Luzin set
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, 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.
The existence of a Luzin space is independent of the axioms of ZFC. Luzin (1914) showed that the continuum hypothesis implies that a Luzin space exists. Kunen (1977) showed that assuming Martin's Axiom and the negation of the continuum hypothesis, there are no Hausdorff Luzin spaces.
In real analysis and descriptive set theory, a Luzin set (or Lusin set), is defined as 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. 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. A weakly Luzin set is an uncountable subset of a real vector space such that for any uncountable subset the set of directions between different elements of the subset is dense in the sphere of directions.
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