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 | 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 .

**[vD]**Eric K. van Douwen,*A basically disconnected normal space Φ with 𝛽Φ-Φ=1*, Canad. J. Math.**31**(1979), no. 5, 911–914. MR**546947**, 10.4153/CJM-1979-086-3**[vD]**-,*Cardinal functions on compact F-spaces and on weakly countably complete Boolean algebras*(to appear).**[VD]**-,*Functions from the integers to the integers and topology*(to appear).**[vDvM]**Eric K. van Douwen and Jan van Mill,*Parovičenko’s characterization of 𝛽𝜔-𝜔 implies CH*, Proc. Amer. Math. Soc.**72**(1978), no. 3, 539–541. MR**509251**, 10.1090/S0002-9939-1978-0509251-7**[GH]**Leonard Gillman and Melvin Henriksen,*Rings of continuous functions in which every finitely generated ideal is principal*, Trans. Amer. Math. Soc.**82**(1956), 366–391. MR**0078980**, 10.1090/S0002-9947-1956-0078980-4**[GJ]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[H]**R. W. Heath,*Separability and ℵ₁-compactness*, Colloq. Math.**12**(1964), 11–14. MR**0167952****[J]**F. B. Jones,*Concerning normal and completely normal spaces*, Bull. Amer. Math. Soc.**43**(1937), no. 10, 671–677. MR**1563615**, 10.1090/S0002-9904-1937-06622-5**[K]**Sabine Koppelberg,*Homomorphic images of 𝜎-complete Boolean algebras*, Proc. Amer. Math. Soc.**51**(1975), 171–175. MR**0376475**, 10.1090/S0002-9939-1975-0376475-9**[L]**Alain Louveau,*Caractérisation des sous-espaces compacts de 𝛽𝑁*, Bull. Sci. Math. (2)**97**(1973), 259–263 (1974) (French). MR**0353261****[MR]**Eric K. van Douwen, J. Donald Monk, and Matatyahu Rubin,*Some questions about Boolean algebras*, Algebra Universalis**11**(1980), no. 2, 220–243. MR**588216**, 10.1007/BF02483101**[P]**C. W. Proctor,*A separable pseudonormal nonmetrizable Moore space*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**18**(1970), 179–181 (English, with Loose Russian summary). MR**0263023**

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Additional Information

DOI:
http://dx.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