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 that 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.
A Luzin set exhibits two fundamental properties: it is nonmeager and has measure zero. The latter follows from the existence of a meager subset within any set of positive measure, leading to the conclusion that the original set is uncountable.