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 nonmeager; that 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, if the continuum hypothesis holds, then every nonmeager set has a Luzin subset.

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 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 has positive outer measure.

References

John C. Oxtoby (1980). Measure and category: a survey of the analogies between topological and measure spaces. Berlin: Springer-Verlag. ISBN 0-387-90508-1.