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

DOI:
https://doi.org/10.1090/S0002-9947-1980-0561827-0

MathSciNet review:
561827

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Abstract | References | Similar Articles | Additional Information

Abstract: Under there is a (compact) strongly zero-dimensional *F*-space of weight which cannot be embedded in any basically disconnected space.

Dually, under there is a weakly countably complete (or almost -complete, or countable separation property) Boolean algebra of cardinality 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 then 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 , the field of subsets of *w*, then there is an embedding such that .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0561827-0

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