Category and Measure
Infinite Combinatorics, Topology and Groups

The continuous structure of topological spaces is examined from the viewpoints of category and measure, with the former being paramount.

N. H. Bingham (Author), Adam J. Ostaszewski (Author)

9780521196079, Cambridge University Press

Hardback, published 23 January 2025

345 pages
23.5 x 15.9 x 2.4 cm, 0.64 kg

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.

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.

Subject Areas: Calculus & mathematical analysis [PBK]