Celebratio Mathematica — Bing

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R. H. Bing, W. W. Bledsoe, and R. D. Mauldin: “Sets generated by rectangles,” Pacific J. Math. 51 : 1 (1974), pp. 27–36. MR 0357124 Zbl 0261.04001 article

For any family \( F \) of sets, let \( \mathcal{B}(F) \) denote the smallest \( \sigma \)-algebra containing \( F \). Throughout this paper \( X \) denotes a set and \( \mathcal{R} \) the family of sets of the form \( A\times B \), for \( A\subseteq X \) and \( B\subseteq X \). It is of interest to find conditions under which the following holds:

(1) Each subset of \( X\times X \) is a member of \( \mathcal{B}(\mathcal{R}) \).

The interesting case is when \( \omega_t < \operatorname{Card} X \leq c \), since results for other cases are known.

It is shown in Theorem 9 that (1) is equivalent to

(2) There is a countable ordinal a such that each subset of \( X\times X \) can be generated from \( \mathcal{R} \) in \( \alpha \) Baire process steps.

It is also shown that the two-dimensional statements (1) and (2) are equivalent to the one-dimensional statement

(3) There is a countable ordinal \( \alpha \) such that for each family \( H \) of subsets of \( X \) with \( \operatorname{Card} H = \operatorname{Card} X \), there is a countable family \( G \) such that each member of \( H \) can be generated from \( G \) in \( \alpha \) steps.

It is shown in Theorem 5 that the continuum hypothesis (CH) is equivalent to certain statements about rectangles of the form (1) and (2) with \( \alpha = 2 \).

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