Abstract
This chapter discusses some peculiar sets of real numbers and some of the methods for obtaining them. Bernstein constructed a set of reals of cardinality the continuum, which is neither disjoint from nor contains an uncountable closed set. His construction used transfinite induction and the fact that every uncountable closed set has cardinality the continuum. A set of reals is meager if it is the countable union of nowhere dense sets. A set of reals is comeager if it is the complement of a meager set. The Baire category theorem says that no complete metric space is meager in itself. Assuming the continuum hypothesis, there is a set of reals of cardinality the continuum that has countable intersection with every measure zero set. A set of reals X has universal measure zero if for all measures μ on the Borel sets, there is a Borel set of μ-measure zero covering X. The existence of uncountable sets of universal measure zero and uncountable perfectly meager sets does not require any axioms beyond the usual Zermelo–Fraenkel with the axiom of choice. There exists a set of reals X of cardinality ω1 which has universal measure zero and is perfectly meager.
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