Sigma^0_alpha - Admissible Representations (Extended Abstract)

Authors Matthew de Brecht, Akihiro Yamamoto

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Matthew de Brecht
Akihiro Yamamoto

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Matthew de Brecht and Akihiro Yamamoto. Sigma^0_alpha - Admissible Representations (Extended Abstract). In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 119-130, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


We investigate a hierarchy of representations of topological spaces by measurable functions that extends the traditional notion of admissible representations common to computable analysis. Specific instances of these representations already occur in the literature (for example, the naive Cauchy representation of the reals and the ``jump'' of a representation), and have been used in investigating the computational properties of discontinuous functions. Our main contribution is the integration of a recently developing descriptive set theory for non-metrizable spaces that allows many previous results to generalize to arbitrary countably based $T_0$ topological spaces. In addition, for a class of topological spaces that include the reals (with the Euclidean topology) and the power set of $\omega$ (with the Scott-topology), we give a complete characterization of the functions that are (topologically) realizable with respect to the level of the representations of the domain and codomain spaces.
  • Admissible representations
  • Borel measurable functions
  • computable analysis
  • descriptive set theory


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