turkmath.org

A subset $A$ of an uncountable $T_2$ space $X$ is called a Luzin Set after the Russian Mathematician Nikolai Luzin iff $|A ∩ N| \leq\aleph_0$ for each nowhere dense subset $N$ of $X$. He proved in 1920 that in the Axiomatic Model (ZFC + CH) there exists a Luzin set in any Euclidean space. The existence of such sets in certain Axiomatic Models of Set Theory then becomes an interesting problem in Set Theoretic Topology . Kenneth Kunen has proved finally in 1976 that this question of existence is actually non- provable in ZFC since in the Model (ZFC+ MA+ ¬CH) such sets does not exist !