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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras

Authors: Eric K. van Douwen and Jan van Mill
Journal: Trans. Amer. Math. Soc. 259 (1980), 121-127
MSC: Primary 54G05; Secondary 03E50, 06E05, 54C15, 54C25
MathSciNet review: 561827
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Abstract: Under $ {\text{MA}}\,{\text{ + }}\,{{\text{2}}^\omega }\, = \,{\omega _2}$ there is a (compact) strongly zero-dimensional F-space of weight $ {2^\omega }$ which cannot be embedded in any basically disconnected space.

Dually, under $ {\text{MA}}\, + \,{2^\omega }\, = \,{\omega _2}$ there is a weakly countably complete (or almost $ \sigma $-complete, or countable separation property) Boolean algebra of cardinality $ {2^\omega }$ which is not a homomorphic image of any countably complete Boolean algebra.

The key to our construction is the observation that if X is a subspace of a basically disconnected space and $ \beta \omega \, \subseteq \,X$ then $ \beta \omega $ is a retract of X.

Dually, if B is a homomorphic image of a countably complete Boolean algebra, and if h is a homomorphism from B onto $ \mathcal{P}(\omega )$, the field of subsets of w, then there is an embedding $ e:\,\mathcal{P}(\omega ) \to \,B$ such that $ h\, \circ \,e\, = \,{\text{i}}{{\text{d}}_{\mathcal{P}(\omega )}}$.

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Additional Information

PII: S 0002-9947(1980)0561827-0
Keywords: Strongly zero-dimensional, Boolean algebra, basically disconnected, countably complete, F-space, weakly countably complete, subspace, quotient, retraction
Article copyright: © Copyright 1980 American Mathematical Society