Luzin space

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.

For continuous images of separable complete metric spaces, known as Lusin spaces, see Polish space ยง Lusin spaces.

The existence of a Luzin space is independent of the axioms of ZFC. Lusin (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.

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