Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Prologue: Regular Variation
- 1 Preliminaries
- 2 Baire Category and Related Results
- 3 Borel Sets, Analytic Sets and Beyond: Δ21
- 4 Infinite Combinatorics in Rn: Shift-Compactness
- 5 Kingman Combinatorics and Shift-Compactness
- 6 Groups and Norms: BirkhoffKakutani Theorem
- 7 Density Topology
- 8 Other Fine Topologies
- 9 CategoryMeasure 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
12 - Continuity and Coincidence Theorems
Published online by Cambridge University Press: 14 January 2025
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Prologue: Regular Variation
- 1 Preliminaries
- 2 Baire Category and Related Results
- 3 Borel Sets, Analytic Sets and Beyond: Δ21
- 4 Infinite Combinatorics in Rn: Shift-Compactness
- 5 Kingman Combinatorics and Shift-Compactness
- 6 Groups and Norms: BirkhoffKakutani Theorem
- 7 Density Topology
- 8 Other Fine Topologies
- 9 CategoryMeasure 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
Summary
The simplest of the continuity theorems considered states that a Baire-measurable function between metric spaces has only a meagre set of discontinuity points. Results on Baire continuity (again, this theme goes back to Banach’s book) are given, for instance the Baire homomorphism theorem states that a Baire homomorphism between normed groups X, Y with X topologically complete is continuous. Another generalization is presented as Banach’s continuous-homomorphism theorem. The coincidence theorems we present derive from Sandro Levi’s 1983 result on the comparison of topologies, to the effect that if one refines the other, they must coincide on a subspace.
Keywords
- Type
- Chapter
- Information
- Category and MeasureInfinite Combinatorics, Topology and Groups, pp. 184 - 202Publisher: Cambridge University PressPrint publication year: 2025