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 also has positive measure. As we all know, except the twat who added the unhelpful 'fact' tag, any set of positive Lebesgue measure contains an unmeasurable set. This unmeasurable set must further be uncountable. Therefore the preceding sentence can be interpreted in two equivalent ways. Either consider that every subset has positive outer measure, or, that every measurable uncountable set has positive Lebesgue 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.