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Enumeration Degrees and Topology

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Sailing Routes in the World of Computation (CiE 2018)

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Abstract

The study of enumeration degrees appears prima facie to be far removed from topology. Work by Miller, and subsequently recent work by Kihara and the author has revealed that actually, there is a strong connection: Substructures of the enumeration degrees correspond to \(\sigma \)-homeomorphism types of second-countable topological spaces. Here, a gentle introduction to the area is attempted.

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Notes

  1. 1.

    Using \(p(k) = n+1\) rather than \(p(k) = n\) is necessary to deal with the empty set in a uniform way.

  2. 2.

    There are some variations here regarding what aspects are required to be effective. Typical names used in the literature are effective topological space, computable topological space or effectively enumerable topological space, see e.g. [25, 45]. These details do not matter for our purposes.

  3. 3.

    We have slightly deviated from the usual definition here. In topology, we would typically demand that the \(\mathbf {X}_i\) can be disjointly embedded into \(\mathbf {Y}\). The difference can be removed by replacing \(\mathbf {Y}\) with \(\mathbf {Y} \times \mathbb {N}\). The results we need from topology hold for either version, and the present one makes the connection to degree theory more elegant.

  4. 4.

    We now realize that this means that there are continuous degrees which are not Turing degrees!

  5. 5.

    Solon uses the name cototal instead of graph-cototal, which we have already used for a different concept above.

  6. 6.

    An enumeration degree is quasi-minimal, if it is non-computable, but every total degree below is computable.

  7. 7.

    This open question was brought to the author’s attention by Joe Miller.

  8. 8.

    This raises the question how exactly one ought to define the Borel hierarchy in these spaces. One approach is found in [36].

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Acknowledgments

My understanding of the subject material has tremendously benefited from a multitude of discussions and talks. Foremost, I am grateful to Takayuki Kihara, but also to Matthew de Brecht, Mathieu Hoyrup, Steffen Lempp, Joseph Miller, Keng Meng Selwyn Ng and Mariya Soskova.

The author received support from the MSCA-RISE project “CID - Computing with Infinite Data” (731143) and the Marie Curie International Research Staff Exchange Scheme Computable Analysis, PIRSES-GA-2011-29496.

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Authors and Affiliations

  1. Swansea University, Swansea, UK

    Arno Pauly

  2. University of Birmingham, Birmingham, UK

    Arno Pauly

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  1. Arno Pauly
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Correspondence to Arno Pauly .

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  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

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Pauly, A. (2018). Enumeration Degrees and Topology. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_33

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  • DOI: https://doi.org/10.1007/978-3-319-94418-0_33

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