Luzin space
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In mathematics, in the realm of topology, a topological space is said to be a Luzin space if it is uncountable and all its nowhere dense subsets are countable.
The real numbers do not form a Luzin space. An example of a nowhere dense uncountable subset of the reals is the Cantor set.
Under the Continuum Hypothesis, there are Luzin spaces, in fact, there are Luzin subsets of the reals. However, assuming Martin's Axiom and the negation of the Continuum Hypothesis, there are no Luzin spaces.
