Computable Closed Euclidean Subsets with and without Computable Points
<|reference_start|>Computable Closed Euclidean Subsets with and without Computable Points: The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are `large' in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality.   This leads us to investigate for various classes of computable real subsets whether they necessarily contain a (not necessarily effectively findable) computable point.<|reference_end|>
@article{roux2006computable,
    title={Computable Closed Euclidean Subsets with and without Computable Points},
    author={St'ephane Le Roux and Martin Ziegler},
    journal={arXiv preprint arXiv:cs/0610080},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0610080},
    primaryClass={cs.LO math.LO}
}