Vol. 51, No. 1, 1974

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Sets generated by rectangles

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

Vol. 51 (1974), No. 1, 27–36

For any family F of sets, let (F) denote the smallest σ-algebra containing F. Throughout this paper X denotes a set and 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 ()

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 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
Received: 14 November 1972
Revised: 12 July 1973
Published: 1 March 1974
R. H. Bing
Woodrow Wilson Bledsoe
R. Daniel Mauldin
Department of Mathematics
University of North Texas
Denton TX 76203-1430
United States