Book contents
- Frontmatter
- Contents
- Introduction
- ERRATA
- Acknowledgements
- 1 Some infinitary combinatorics
- 2 Introducing the chain conditions
- 3 Chain conditions in products
- 4 Classes of calibres, using Σ-products
- 5 Calibres of compact spaces
- 6 Strictly positive measures
- 7 Between property (K) and the countable chain condition
- 8 Classes of compact-calibres, using spaces of ultrafilters
- 9 Pseudo-compactness numbers: examples
- 10 Continuous functions on product spaces
- Appendix: preliminaries
- References
- Subject index
- Index of symbols
7 - Between property (K) and the countable chain condition
Published online by Cambridge University Press: 05 November 2011
- Frontmatter
- Contents
- Introduction
- ERRATA
- Acknowledgements
- 1 Some infinitary combinatorics
- 2 Introducing the chain conditions
- 3 Chain conditions in products
- 4 Classes of calibres, using Σ-products
- 5 Calibres of compact spaces
- 6 Strictly positive measures
- 7 Between property (K) and the countable chain condition
- 8 Classes of compact-calibres, using spaces of ultrafilters
- 9 Pseudo-compactness numbers: examples
- 10 Continuous functions on product spaces
- Appendix: preliminaries
- References
- Subject index
- Index of symbols
Summary
We have seen in Chapter 6 that the c.c.c. property (for compact spaces) branches into two stronger and logically independent properties: the property of calibre ω+ and the existence of a strictly positive measure. These two properties have in fact a common denominator, stronger than c.c.c: Knaster's property (K) (indeed, property Kn for 2 ≤ n < ω). Between the c.c.c. and property (K) there is a significant qualitative difference: the c.c.c. cannot be proved in ZFC to be productive, but property (K) is productive (Theorem 2.2(a)).
In this chapter we are concerned with differentiating property (K) from c.c.c, and with properties that lie between the two. The principal results, both assuming the continuum hypothesis and using combinatorial methods, are these:
(a) There is a c.c.c space X such that X × X is not a c.c.c space (Theorem 7.13, due to R. Laver and F. Galvin); and
(b) there is a productively c.c.c. space that does not have property (K) (Theorem 7.9, due to K. Kunen).
In the Notes to this chapter we describe the effect of Martin's axiom on the countable chain properties.
Kunen's example
7.1 Lemma. Let α be an infinite cardinal and X a space such that S(X) ≤ cf(α). If {Vξ : ξ, < α} is a set of non-empty open subsets of X such that Vξ′ ⊂ Vξ for ξ < ξ′ < α, then {cl Vξ : ξ < α} stabilizes.
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- Information
- Chain Conditions in Topology , pp. 180 - 207Publisher: Cambridge University PressPrint publication year: 1982