Abstract
We discuss ideal presentations of effective quasi-Polish spaces and some of their subclasses. Based on this, we introduce and study natural numberings of these classes, in analogy with the numberings of classes of algebraic structures popular in computability theory. We estimate the complexity of (effective) homeomorphism w.r.t. these numberings, and of some natural index sets. In particular, we give precise characterizations of the complexity of certain classes related to separation axioms.
M. de Brecht—De Brecht’s research was supported by JSPS KAKENHI Grant Number 18K11166.
T. Kihara—Kihara’s research was partially supported by JSPS KAKENHI Grant Numbers 19K03602 and 21H03392, and the JSPS-RFBR Bilateral Joint Research Project JPJSBP120204809.
V. Selivanov—Selivanov’s research was supported by the RFBR-JSPS Bilateral Joint Research Project 20-51-50001.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
We thank Nikolay Bazhenov for the related bibliographical hints.
References
Abramsky S., Jung, A.: Domain theory. In: Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford (1994)
Ash, C., Knight, J.: Computable Structures and the Hyperarithmetical Hierarchy. Studies in Logic and the Foundations of Mathematics, vol. 144. North-Holland Publishing Co., Amsterdam (2000)
de Brecht, M.: Quasi-Polish spaces. Ann. Pure Appl. Logic 164, 356–381 (2013)
de Brecht, M.: Some notes on spaces of ideals and computable topology. In: Anselmo, M., Della Vedova, G., Manea, F., Pauly, A. (eds.) CiE 2020. LNCS, vol. 12098, pp. 26–37. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51466-2_3
Downey R., Melnikov A.G.: Effectively compact spaces. Unpublished manuscript
de Brecht, M., Pauly, A., Schröder, M.: Overt choice. Computability 9(3–4), 169–191 (2020)
Ershov, Y.L., Goncharov, S.S.: Constructive Models. Plenum, New York (1999)
Fokina, E.D., Friedman, S.D., Harizanov, V., Knight, J.F., McCoy, C., Montalban, A.: Isomrphism relations on computable structures. J. Symb. Log. 77(1), 122–132 (2012)
Fokina, E., Friedman, S., Nies, A.: Equivalence relations that are \(\Sigma ^0_3\) complete for computable reducibility. In: Ong, L., de Queiroz, R. (eds.) WoLLIC 2012. LNCS, vol. 7456, pp. 26–33. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32621-9_2
Fokina, E.B., Harizanov, V., Melnikov, A.: Computable model theory. In: Turing’s Legacy: Developments From Turing’s Ideas in Logic. Lecture Notes Logic, vol. 42, pp. 124–194. Association for Symbolic Logic, La Jolla (2014)
Goncharov S.S., Knight J.: Computable structure and antistructure theorems. Algebra Logika 41(6), 639–681, 757 (2002)
Hoyrup, M., Kihara, T., Selivanov, V. Degree spectra of homeomorphism types of Polish spaces. Arxiv 2004.06872v1 (2020)
Hoyrup, M., Rojas, C., Selivanov, V., Stull, D.M.: Computability on Quasi-Polish spaces. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds.) DCFS 2019. LNCS, vol. 11612, pp. 171–183. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-23247-4_13
Harrison-Trainor M., Melnikov A., Ng K.M.: Computability of Polish spaces up to homeomorphism. J. Symb. Log. 85(4), 1–25 (2020)
Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995). https://doi.org/10.1007/978-1-4612-4190-4
Korovina, M., Kudinov, O.: On higher effective descriptive set theory. In: Kari, J., Manea, F., Petre, I. (eds.) CiE 2017. LNCS, vol. 10307, pp. 282–291. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58741-7_27
Kihara, T., Ng, K.M., Pauly, A.: Enumeration degrees and nonmetrizable topology. Submitted. arXiv:1904.04107
McNicholl, T.H., Stull, D.M.: The isometry degree of a computable copy of \(\ell ^{p1}\). Computability 8(2), 179–189 (2019)
Melnikov, A.G.: Computably isometric spaces. J. Symb. Log. 78(4), 1055–1085 (2013)
Selivanov, V.L.: Positive structures. In: Barry Cooper, B., Goncharov, S.S. (eds.) Computability and Models, Perspectives East and West, pp. 321–350. Kluwer Academic/Plenum Publishers, New York (2003)
Selivanov, V.L.: Towards the effective descriptive set theory. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 324–333. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20028-6_33
Selivanov, V.: On degree spectra of topological spaces. Lobachevskii J. Math. 41(2), 252–259 (2020)
Steen, L.A., Seebach, J.A.: Counterexamples in Topology. Springer, New York (1978). https://doi.org/10.1007/978-1-4612-6290-9. Reprinted by Dover Publications, New York, 1995
Acknowledgement
The authors are grateful to anonymous referees for the careful reading and valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
de Brecht, M., Kihara, T., Selivanov, V. (2022). Enumerating Classes of Effective Quasi-Polish Spaces. In: Berger, U., Franklin, J.N.Y., Manea, F., Pauly, A. (eds) Revolutions and Revelations in Computability. CiE 2022. Lecture Notes in Computer Science, vol 13359. Springer, Cham. https://doi.org/10.1007/978-3-031-08740-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-08740-0_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-08739-4
Online ISBN: 978-3-031-08740-0
eBook Packages: Computer ScienceComputer Science (R0)