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Local Compactness for Computable Polish Metric Spaces is \(\varPi ^1_1\)-complete

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Evolving Computability (CiE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9136))

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Abstract

We show that the property of being locally compact for computable Polish metric spaces is \(\varPi ^1_1\) complete. We verify that local compactness for Polish metric spaces can be expressed by a sentence in \(L_{\omega _1, \omega }\).

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References

  1. Kechris, A.S.: Classical descriptive set theory, vol. 156. Springer, New York (1995)

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  2. Melnikov, A.G., Nies, A.: The Classification Problem for Compact Computable Metric Spaces. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 320–328. Springer, Heidelberg (2013)

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  3. Naulin, R., Aylwin, C.: On the complexity of the family of compact subsets of \(\mathbb{Q}\). Notas de Mat. 5(2), 283 (2009)

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Acknowledgment

This work was carried out at the Hausdorff Institute for Mathematics in October 2013, and at the Research Centre Whiritoa in December 2014.

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Authors and Affiliations

  1. Department of Computer Science, University of Auckland, Auckland, New Zealand

    André Nies

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Champaign, IL, USA

    Slawomir Solecki

Authors
  1. André Nies
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  2. Slawomir Solecki
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Corresponding author

Correspondence to André Nies .

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Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. University of Bucharest, Bucharest, Romania

    Victor Mitrana

  3. Sofia University, Sofia, Bulgaria

    Mariya Soskova

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© 2015 Springer International Publishing Switzerland

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Nies, A., Solecki, S. (2015). Local Compactness for Computable Polish Metric Spaces is \(\varPi ^1_1\)-complete. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_29

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  • DOI: https://doi.org/10.1007/978-3-319-20028-6_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20027-9

  • Online ISBN: 978-3-319-20028-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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