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
10 - Continuous functions on product spaces
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 begin this chapter with a proof that under suitable hypotheses a function f continuous on a product space XI (or even on some of its dense subspaces) is effectively defined on a ‘small’ subproduct (in the sense that there are ‘small’ J ⊂ I and continuous g on XJ such that f = g°πJ). We apply this result, together with some of the product-space theorems of Chapter 3, to obtain an identification in concrete form of the Stone–Čech compactification, the Hewitt realcompactification, and the Dieudonné topological completion of certain completely regular Hausdorff spaces (these terms are defined in Appendix B). Such an opportunity occurs infrequently since the Stone–Čech compactification of a space X, whose elements are normally described as maximal filters in the class of zero-sets of X, is defined by an appeal to (a variant of) Zorn's lemma; it does not arise in practice as a readily identifiable, familiar space.
Among the specific results proved are these (cf. Corollary 10.7 and Theorem 10.17): in a product of metrizable spaces, and in a product of compact spaces, every ω+-Σ-space is C-embedded. We conclude with an example, one of several due to Ulmer, showing that there do exist ω+-Σ-products not C-embedded in XI.
Definition. Let Y ⊂ XI, J ⊂ I, let Z be a set and f : Y → Z.
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- Information
- Chain Conditions in Topology , pp. 231 - 250Publisher: Cambridge University PressPrint publication year: 1982