Category and Measure

Infinite Combinatorics, Topology and Groups

N. H. Bingham, Adam J. Ostaszewski

• Publisher: Cambridge University Press
• Year: 2025
• eBook ISBN: 9781009640022
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

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.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Imprints page
• Dedication
• Contents
• Preface
• Prologue: Regular Variation
• 1 Preliminaries
• 2 Baire Category and Related Results
• 3 Borel Sets, Analytic Sets and Beyond: delta[sup(1)][sub(2)]
• 4 Infinite Combinatorics in R[sup(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