Bernstein set

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

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