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2 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.

  • ON THE DENSITY TYPE TOPOLOGIES IN HIGHER DIMENSIONS

  • Part of
  • GRAŻYNA HORBACZEWSKA
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 83 / Issue 1 / February 2011
    Published online by Cambridge University Press:
    14 September 2010, pp. 158-170
    Print publication:
    February 2011
    • Article
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  • The topologies of the density type in Euclidean space of dimension higher than one are introduced. Definitions are based on a notion of density point connected with a set of sequences of real numbers. Our purpose is to study properties of these topologies and connections between them.