Sets generated by rectangles

R. H. Bing, Woodrow Wilson Bledsoe and R. Daniel Mauldin

Vol. 51 (1974), No. 1, 27–36

DOI: 10.2140/pjm.1974.51.27

Abstract

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

(1) Each subset of X × X is a member of ℬ(ℛ₋)

The interesting case is when ω₁ < Card X ≦ 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 α such that each subset of X × X oan be generated from ℛ is α 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 α such that for each family H of subsets of X with Card H = Card X, there is a oountable family G such that each member of H can be generated from G in α 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 α = 2.

Mathematical Subject Classification 2000

Primary: 04A15

Secondary: 28A05, 04A30, 04A20

Milestones

Received: 14 November 1972

Revised: 12 July 1973

Published: 1 March 1974