# On character and chain conditions in images of products

Fundamenta Mathematicae (1998)

• Volume: 158, Issue: 1, page 41-49
• ISSN: 0016-2736

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## Abstract

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A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property ${R}_{\lambda }^{\text{'}}$ which we show is satisfied by all ξ-adic spaces. Whereas Property ${R}_{\lambda }^{\text{'}}$ is productive, we show that a weaker (but more natural) Property ${R}_{\lambda }$ is not productive. Polyadic spaces are shown to satisfy a stronger chain condition called Property ${R}_{\lambda }^{\text{'}\text{'}}$. We use Property ${R}_{\lambda }^{\text{'}}$ to show that not all compact, monolithic, scattered spaces are ξ-adic, thus answering a question of Chertanov’s.

## How to cite

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Bell, Murray. "On character and chain conditions in images of products." Fundamenta Mathematicae 158.1 (1998): 41-49. <http://eudml.org/doc/212301>.

@article{Bell1998,
abstract = {A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property $R_λ^\{\prime \}$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_λ^\{\prime \}$ is productive, we show that a weaker (but more natural) Property $R_λ$ is not productive. Polyadic spaces are shown to satisfy a stronger chain condition called Property $R_λ^\{\prime \prime \}$. We use Property $R_λ^\{\prime \}$ to show that not all compact, monolithic, scattered spaces are ξ-adic, thus answering a question of Chertanov’s.},
author = {Bell, Murray},
journal = {Fundamenta Mathematicae},
keywords = {compact; scattered; products; chain condition; ordinals; product of compact scattered spaces; weight; character; dyadic spaces},
language = {eng},
number = {1},
pages = {41-49},
title = {On character and chain conditions in images of products},
url = {http://eudml.org/doc/212301},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Bell, Murray
TI - On character and chain conditions in images of products
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 1
SP - 41
EP - 49
AB - A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property $R_λ^{\prime }$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_λ^{\prime }$ is productive, we show that a weaker (but more natural) Property $R_λ$ is not productive. Polyadic spaces are shown to satisfy a stronger chain condition called Property $R_λ^{\prime \prime }$. We use Property $R_λ^{\prime }$ to show that not all compact, monolithic, scattered spaces are ξ-adic, thus answering a question of Chertanov’s.
LA - eng
KW - compact; scattered; products; chain condition; ordinals; product of compact scattered spaces; weight; character; dyadic spaces
UR - http://eudml.org/doc/212301
ER -

## References

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3. [Ch88] G. Chertanov, z Continuous images of products of scattered compact spaces, Siberian Math. J. 29 (1988), no. 6, 1005-1012. Zbl0725.54021
4. [EHMR84] P. Erdős, A. Hajnal, A. Máté and R. Rado, z Combinatorial Set Theory: Partition Relations for Cardinals, Stud. Logic Found. Math. 106, North-Holland, 1984. Zbl0573.03019
5. [Ge73] J. Gerlits, z On a problem of S. Mrówka, Period. Math. Hungar. 4 (1973), no. 1, 71-79.
6. [Ho84] R. Hodel, z Cardinal functions I, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, 1984, 1-61.
7. [HBA89] S. Koppelberg, z Handbook of Boolean Algebras, Vol. 1, J. D. Monk and R. Bonnet (eds.), North-Holland, 1989.
8. [Mr70] S. Mrówka, z Mazur theorem and $m$-adic spaces, Bull. Acad. Polon. Sci. 18 (1970), no. 6, 299-305. Zbl0194.54302

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