A Note on Closed Subsets in Quasi-zero-dimensional Qcb-spaces (Extended Abstract)

Author Matthias Schröder

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Matthias Schröder

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Matthias Schröder. A Note on Closed Subsets in Quasi-zero-dimensional Qcb-spaces (Extended Abstract). In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 233-244, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


We introduce the notion of quasi-zero-dimensionality as a substitute for the notion of zero-dimensionality, motivated by the fact that the latter behaves badly in the realm of qcb-spaces. We prove that the category $\QZ$ of quasi-zero-dimensional qcb$_0$-spaces is cartesian closed. Prominent examples of spaces in $\QZ$ are the spaces in the sequential hierarchy of the Kleene-Kreisel continuous functionals. Moreover, we characterise some types of closed subsets of $\QZ$-spaces in terms of their ability to allow extendability of continuous functions. These results are related to an open problem in Computable Analysis.
  • Computable analysis
  • Qcb-spaces
  • extendability


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