# Sh:561

• Shelah, S., & Zapletal, J. (1997). Embeddings of Cohen algebras. Adv. Math., 126(2), 93–118.
• Abstract:
Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(\kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space \{0,1\}^\kappa equipped with product measure, and C(\kappa), the algebra of regular open sets in the space \{0,1\}^\kappa, for \kappa an infinite cardinal. C(\kappa) is much easier to analyse than B(\kappa): C(\kappa) has a dense subset of size \kappa, while the density of B(\kappa) depends on the cardinal characteristics of the real line; and the definition of C(\kappa) is simpler. Indeed, C(\kappa) seems to have the simplest definition among all algebras of its size. In the Main Theorem of this paper we show that in a certain precise sense, C(\aleph_1) has the simplest structure among all algebras of its size, too.

MAIN THEOREM: If ZFC is consistent then so is ZFC + 2^{\aleph_0}=\aleph_2 +“for every complete Boolean algebra B of uniform density \aleph_1, C(\aleph_1) is isomorphic to a complete subalgebra of B”.

• Version 1995-02-14_10 (19p) published version (26p)
Bib entry
@article{Sh:561,
author = {Shelah, Saharon and Zapletal, Jind{\v{r}}ich},
title = {{Embeddings of Cohen algebras}},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {126},
number = {2},
year = {1997},
pages = {93--118},
issn = {0001-8708},
mrnumber = {1442306},
mrclass = {03E35 (03E05 03E40 03E50 03G05 06E10)},
doi = {10.1006/aima.1996.1597},
note = {\href{https://arxiv.org/abs/math/9502230}{arXiv: math/9502230}},
arxiv_number = {math/9502230}
}