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Full Description
Topological spaces in general, and the real numbers in particular, have the characteristic of exhibiting a 'continuity structure', one that can be examined from the vantage point of Baire category or of Lebesgue measure. Though they are in some sense dual, work over the last half-century has shown that it is the former, topological view, that has pride of place since it reveals a much richer structure that draws from, and gives back to, areas such as analytic sets, infinite games, probability, infinite combinatorics, descriptive set theory and topology. Keeping prerequisites to a minimum, the authors provide a new exposition and synthesis of the extensive mathematical theory needed to understand the subject's current state of knowledge, and they complement their presentation with a thorough bibliography of source material and pointers to further work. The result is a book that will be the standard reference for all researchers in the area.
Contents
Prologue. Regular variation; 1. Preliminaries; 2. Baire category and related results; 3. Borel sets, analytic sets and beyond: $\Delta^1_2$; 4. Infinite combinatorics in $\mathbb{R}^n$: shift-compactness; 5. Kingman combinatorics and shift-compactness; 6. Groups and norms: Birkhoff-Kakutani theorem; 7. Density topology; 8. Other fine topologies; 9. Category-measure duality; 10. Category embedding theorem and infinite combinatorics; 11. Effros' theorem and the cornerstone theorems of functional analysis; 12. Continuity and coincidence theorems; 13. * Non-separable variants; 14. Contrasts between category and measure; 15. Interior point theorems: Steinhaus-Weil theory; 16. Axiomatics of set theory; Epilogue. Topological regular variation; References; Index.
