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1 results

  • 12 - Continuity and Coincidence Theorems

  • 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 184-202

  • 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.