Skip to main content
SpringerLink
Log in

Computable approximations of a chainable continuum with a computable endpoint

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

It is known that a semicomputable continuum S in a computable topological space can be approximated by a computable subcontinuum by any given precision under condition that S is chainable and decomposable. In this paper we show that decomposability can be replaced by the assumption that S is chainable from a to b, where a is a computable point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Specker, E.: Der satz vom maximum in der rekursiven analysis. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 254–265. North Holland Publ. Comp., Amsterdam (1959)

    Google Scholar 

  2. Miller, J.S.: Effectiveness for embedded spheres and balls. Electr. Notes Theorit. Comput. Sci. 66, 127–138 (2002)

    Article  Google Scholar 

  3. Iljazović, Z.: Co-c.e. spheres and cells in computable metric spaces. Log. Methods Comput. Sci. 7(3:05), 1–21 (2011)

    MathSciNet  Google Scholar 

  4. Iljazović, Z.: Compact manifolds with computable boundaries. Log. Methods Comput. Sci. 9(4:19), 1–22 (2013)

    MathSciNet  Google Scholar 

  5. Iljazović, Z., Sušić, I.: Semicomputable manifolds in computable topological spaces. J. Complex. 45, 83–114 (2018)

    Article  MathSciNet  Google Scholar 

  6. Burnik, K., Iljazović, Z.: Computability of 1-manifolds. Log. Methods Comput. Sci. 10(2:8), 1–28 (2014)

    MathSciNet  Google Scholar 

  7. Brattka, V.: Plottable real number functions and the computable graph theorem. SIAM J. Comput. 38(1), 303–328 (2008)

    Article  MathSciNet  Google Scholar 

  8. Iljazović, Z., Pažek, B.: Co-c.e. sets with disconnected complements. Theory Comput. Syst. 62(5), 1109–1124 (2018)

    Article  MathSciNet  Google Scholar 

  9. Čelar, M., Iljazović, Z.: Computability of products of chainable continua. Theory Comput. Syst. 65, 410–427 (2021)

    Article  MathSciNet  Google Scholar 

  10. Čelar, M., Iljazović, Z.: Computability of glued manifolds. J. Log. Comput. 32(1), 65–97 (2022)

    Article  MathSciNet  Google Scholar 

  11. Iljazović, Z.: Computability of graphs. Math. Log. Q. 66(1), 51–64 (2020)

    Article  MathSciNet  Google Scholar 

  12. Amir, D.E., Hoyrup, M.: Computability of finite simplicial complexes. In: Bojańczyk, M., et al., (eds), 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), pp. 111:1–16. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany, (2022)

  13. Kihara, T.: Incomputability of simply connected planar continua. Computability 1(2), 131–152 (2012)

    Article  MathSciNet  Google Scholar 

  14. Tanaka, H.: On a \(\Pi _{1}^{0}\) set of positive measure. Nagoya Math. J. 38, 139–144 (1970)

    Article  MathSciNet  Google Scholar 

  15. Hertling, P.: Is the Mandelbrot set computable? Math. Log. Q. 51, 5–18 (2005)

    Article  MathSciNet  Google Scholar 

  16. Iljazović, Z.: Chainable and circularly chainable co-c.e. sets in computable metric spaces. J. Univ. Comput. Sci. 15(6), 1206–1235 (2009)

    MathSciNet  Google Scholar 

  17. Iljazović, Z., Pažek, B.: Computable intersection points. Computability 7, 57–99 (2018)

    Article  MathSciNet  Google Scholar 

  18. Čačić, V., Horvat, M., Iljazović, Z.: Computable subcontinua of semicomputable chainable hausdorff continua. Theoret. Comput. Sci. 892, 155–169 (2021)

    Article  MathSciNet  Google Scholar 

  19. Nadler, S.B.: Continuum Theory. Marcel Dekker Inc., New York (1992)

    Google Scholar 

  20. Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1989)

    Book  Google Scholar 

  21. Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)

    Book  Google Scholar 

  22. Turing, A.M.: On computable numbers, with an application to the entscheidungsproblem. Proc. London Math. Soc. 42, 230–265 (1936)

    MathSciNet  Google Scholar 

  23. Weihrauch, K.: Computability on computable metric spaces. Theoret. Comput. Sci. 113, 191–210 (1993)

    Article  MathSciNet  Google Scholar 

  24. Brattka, V., Weihrauch, K.: Computability on subsets of Euclidean space I: Closed and compact subsets. Theoret. Comput. Sci. 219, 65–93 (1999)

    Article  MathSciNet  Google Scholar 

  25. Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theoret. Comput. Sci. 305, 43–76 (2003)

    Article  MathSciNet  Google Scholar 

  26. Weihrauch, K.: Computable separation in topology, from \(T_{0}\) to \(T_{2}\). J. Univ. Comput. Sci. 16(18), 2733–2753 (2010)

    Google Scholar 

  27. Weihrauch, K., Grubba, T.: Elementary computable topology. J. Univ. Comput. Sci. 15(6), 1381–1422 (2009)

    MathSciNet  Google Scholar 

  28. Čičković, E., Iljazović, Z., Validžić, L.: Chainable and circularly chainable semicomputable sets in computable topological spaces. Arch. Math. Logic 58, 885–897 (2019)

    Article  MathSciNet  Google Scholar 

  29. Christenson, C.O., Voxman, W.L.: Aspects of Topology. Marcel Dekker Inc., New York (1977)

    Google Scholar 

Download references

Funding

This work was fully supported by the Croatian Science Foundation under the project 7459 CompStruct. This research is partially supported through project KK.01.1.1.02.0027, a project co-financed by the Croatian Government and the European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme.

Author information

Author notes

  1. Zvonko Iljazović and Matea Jelić have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, Zagreb, 10000, Croatia

    Zvonko Iljazović

  2. Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Matice hrvatske 15, Split, 21000, Croatia

    Matea Jelić

Authors
  1. Zvonko Iljazović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matea Jelić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zvonko Iljazović.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iljazović, Z., Jelić, M. Computable approximations of a chainable continuum with a computable endpoint. Arch. Math. Logic 63, 181–201 (2024). https://doi.org/10.1007/s00153-023-00891-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-023-00891-5

Keywords

Mathematics Subject Classification

Search

Navigation