Luzin space

In real analysis and descriptive set theory, a Luzin set (also Lusin set) is an uncountable set A such that every uncountable subset of A is of second Baire category. Equivalently, A is an uncountable set which meets every first category set in only countably many points. Luzin proved that such a set exists as a subset of every second category set. The proof requires the continuum hypothesis.

Obvious properties of a Luzin set are that it must be itself of second category (otherwise the set itself is an uncountable first category subset) and of measure zero, because every set of positive contains a first category set which also has positive measure, and is therefore uncountable.

The measure-category duality provides a measure analogue of Luzin sets - sets of positive measure every uncountable subset of which also has positive measure.