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

↑ 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, https://books.google.com/books?id=WwmvxtDlz9UC&pg=PA163 .

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[oxtoby-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, https://books.google.com/books?id=WwmvxtDlz9UC&pg=PA163 .