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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9136)

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 \(L_{\omega _1, \omega }\).

Notes

Acknowledgment

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

References

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

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignChampaignUSA

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