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1 results

  • 2 - Baire Category and Related Results

  • N. H. Bingham, Imperial College London, Adam J. Ostaszewski, London School of Economics and Political Science
  • Book:
    Category and Measure
    Published online:
    14 January 2025
    Print publication:
    23 January 2025, pp 34-50

  • Summary

    As well as BGT, the other main influence on this book is Oxtoby’s Measure and Category: A Survey of the Analogies between Topological and Measures Spaces (Springer, 1971). For Oxtoby, (Lebesgue) measure is primary, (Baire) category is secondary. Our view, as our title shows, reverses this. The book may thus be regarded as an extended demonstration of the power and wide applicability of the Baire category theorem. Chapter 2 – where we use ‘meagre’ and ‘non-meagre’ for ‘of first (Baire) category’ and ‘of second category’ – proves and discusses several versions of Baire’s (category) theorem: on the line, the intersection of any sequence of dense open sets is dense. We also discuss Baire measurability, and the Baire property. We likewise give a full treatment of the Banach category theorem – a union of any family of meagre open sets is meagre – also used extensively in the book. We discuss countability conditions, and games of Banach–Mazur type. The chapter ends with a discussion of p-spaces (plumed spaces).