In mathematics, a **Bernstein set** is a subset of the real line that meets every uncountable closed subset of the real line but that contains none of them.^{[1]}

A Bernstein set partitions the real line into two pieces in a peculiar way: every measurable set of positive measure meets both the Bernstein set and its complement, as does every set with the property of Baire that is not a meagre set.^{[2]}

## References

**^**Oxtoby, John C. (1980).*Measure and Category*(2nd ed.). p. 24.**^**Morgan, John C., II (1989),*Point Set Theory*, Chapman & Hall/CRC Pure and Applied Mathematics,**131**, CRC Press, p. 163, ISBN 9780824781781.

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