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- W1601632810 abstract "Let M be a set (in this note, M will denote the set of real numbers), and to each x E M, let there correspond a set S(x) c M, called the PictUYf? of s, such that x $ S(x). A subset M’ of M is called independent (or free) if, for each pair of points x and y in M, x $ S(y) and y $ S(x). In [2, I, p. 521 it was conjectured that if M is the set of real numbers, and if the measure of S(x) is bounded and S(x) is not everywhere dense, then there always exists an independent pair. In fact, it is easy to see that if we assume c = N,, then this conjecture is false. To construct a counterexample, we well-order M into an W, -sequence { x~} (a, < a,). For each (Y, we write" @default.
- W1601632810 created "2016-06-24" @default.
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- W1601632810 date "1953-01-01" @default.
- W1601632810 modified "2023-10-16" @default.
- W1601632810 title "Some remarks on set theory. III." @default.
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- W1601632810 doi "https://doi.org/10.1307/mmj/1028989869" @default.
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