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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ K$-triviality in computable metric spaces


Authors: Alexander Melnikov and André Nies
Journal: Proc. Amer. Math. Soc. 141 (2013), 2885-2899
MSC (2010): Primary 03D32, 03F60
Published electronically: April 4, 2013
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Abstract: A point $ x$ in a computable metric space is called $ K$-trivial if
for each positive rational $ \delta $ there is an approximation $ p$ at distance at most $ \delta $ from $ x$ such that the pair $ p, \delta $ is highly compressible in the sense that $ K(p, \delta ) \le K(\delta ) + O(1)$. We show that this local definition is equivalent to the point having a Cauchy name that is $ K$-trivial when viewed as a function from $ \mathbb{N}$ to $ \mathbb{N}$. We use this to transfer known results on $ K$-triviality for functions to the more general setting of metric spaces. For instance, we show that each computable Polish space without isolated points contains an incomputable
$ K$-trivial point.


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Additional Information

Alexander Melnikov
Affiliation: Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand
Address at time of publication: Department of Mathematics, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798

André Nies
Affiliation: Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: andre@cs.auckland.ac.nz

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11528-5
PII: S 0002-9939(2013)11528-5
Keywords: Computable analysis, metric spaces, $K$-triviality
Received by editor(s): October 18, 2011
Received by editor(s) in revised form: October 28, 2011
Published electronically: April 4, 2013
Additional Notes: Both authors were partially supported by the Marsden Fund of New Zealand, grant No. 08-UOA-187.
Communicated by: Julia Knight
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.