Home  ›  Continuity Completion of Preimage is Included in Completion of

Continuity Completion of Preimage is Included in Completion of

Written By Wednesday, October 19, 2022 Add Comment Edit

Abstract

In this paper we introduce the notions of continuity, meet continuity and quasicontinuity of Δ-spaces, where a Δ-space is a monotone determined, weak monotone convergence space. We prove that: a Δ-space \((X, \mathcal {O}(X))\) is continuous iff the topology lattice \(\mathcal {O}(X)\) is completely distributive; a Δ-space \((X, \mathcal {O}(X))\) is meet continuous iff \(\mathcal {O}(X)\) is a complete Heyting algebra; a Δ-space \((X, \mathcal {O}(X))\) is continuous iff it is both meet continuous and quasicontinuous. The concepts of continuity, s 2-continuity, 𝜃-continuity, strong continuity of posets are shown to be special cases of the continuity of Δ-spaces. We also introduce a type of directed completion of Δ-spaces, called Scott completion. For each Δ-space \((X, \mathcal {O}(X))\), the topology lattice \(\mathcal {O}(X)\) is isomorphic to the topology lattice of its Scott completion. The D-completion, D 𝜃 -completion and \(D_{s_{2}}\)-completion of posets are included in the Scott completion.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Indonesia)

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Andradi, H., Ho, W.K.: On a new convergence class in sup-sober spaces. arXiv:1709.03269 (2017)

  2. Erné, M.: Scott convergence and Scott topology in partially ordered sets II. In: Banaschewski, B., Hoffmann, R.-E. (eds.) Continuous Lattices Proceedings, Bremen 1979, vol. 871, pp 61–96. Springer, Berlin (1981)

  3. Erné, M.: Lattice representations for categories of closure spaces. In: Bentley, H.L. et al. (eds.) Categorical Topology, Proc. Conf. Toledo, Ohio, Hldermann. Berlin, pp 197–222 (1983)

  4. Erné, M.: Minimal bases, ideal extensions, and basic dualities. Topology Proc. 29, 445–489 (2005)

    MathSciNet  MATH  Google Scholar

  5. Erné, M.: Infinite distributive laws versus local connectedness and compactness properties. Topol. Appl. 156(12), 2054–2069 (2009)

    Article  MathSciNet  Google Scholar

  6. Erné, M.: Categories of locally hypercompact spaces and quasicontinuous posets. Appl. Categ. Struct. 26(5), 823–854 (2018)

    Article  MathSciNet  Google Scholar

  7. Erné, M.: Web spaces and worldwide web spaces: topological aspects of domain theory. Log. Methods Comput. Sci. 15(1), 23:1–23:38 (2019)

    MathSciNet  MATH  Google Scholar

  8. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar

  9. Gierz, G., Lawson, J. D., Stralka, A.: Quasicontinuous posets. Houst. J. Math. 9, 191–208 (1983)

    MathSciNet  MATH  Google Scholar

  10. González, L.J., Jansana, R.: A topological duality for posets. Algebra Universalis 76, 1–24 (2016)

    Article  MathSciNet  Google Scholar

  11. Hoffmann, R.-E.: Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications. In: Banaschewski, B., Hoffmann, R.-E. (eds.) Continuous Lattices Proceedings, Bremen 1979, vol. 871, pp 159–208. Springer, Berlin (1981)

  12. Keimel, K., Lawson, J.D.: D-completions and the d-topology. Ann Pure Appl Logic 159(3), 292–306 (2009)

    Article  MathSciNet  Google Scholar

  13. Kou, H., Liu, Y., Luo, M.: On meet-continuous dcpos. In: Zhang, G.Q., Lawson, J., Liu, Y.M., Luo, M.K. (eds.) Domain Theory, Logic and Computation, pp 137–149. Kluwer Academic Publishers (2003)

  14. Lawson, J. D.: The duality of continuous posets. Houst. J. Math. 5(3), 357–386 (1979)

    MathSciNet  MATH  Google Scholar

  15. Lawson, J.D.: T 0-spaces and pointwise convergence. Topol. Appl. 21(1), 73–76 (1985)

    Article  Google Scholar

  16. Mao, X., Xu, L.: Quasicontinuity of posets via Scott topology and sobrification. Order 23(4), 359–369 (2006)

    Article  MathSciNet  Google Scholar

  17. Mao, X., Xu, L.: Meet continuity properties of posets. Theor. Comput. Sci. 410(42), 4234–4240 (2009)

    Article  MathSciNet  Google Scholar

  18. Xu, L., Mao, X.: Strongly continuous posets and the local Scott topology. J. Math. Anal. Appl. 345(2), 816–824 (2008)

    Article  MathSciNet  Google Scholar

  19. Xu, X., Yang, J.: Topological representations of distributive hypercontinuous lattices. Chinese Ann. Math. Series B 30(2), 199–206 (2009)

    Article  MathSciNet  Google Scholar

  20. Zhang, W., Xu, X.: s 2-Quasicontinuous posets. Theor. Comput. Sci. 574, 78–85 (2015)

    Article  Google Scholar

  21. Zhang, Z., Li, Q., Jia, X.: 𝜃-Continuity and D 𝜃 -completion of posets. Math. Struct. Comput. Sci. 28 (4), 533–547 (2018). https://doi.org/10.1017/S0960129517000020

    Article  MathSciNet  Google Scholar

  22. Zhao, D., Fan, T.: Dcpo-completion of posets. Theor. Comput. Sci. 411(22), 2167–2173 (2010)

    Article  MathSciNet  Google Scholar

  23. Zhao, D., Ho, W.K.: On topologies defined by irreducible sets. J. Log. Algebraic Methods Program. 84(1), 185–195 (2015)

    Article  MathSciNet  Google Scholar

Download references

Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos.11801491, 11871097, 11771134, 11971448), and Shandong Provincial Natural Science Foundation, China (No. ZR2018BA004). We would like to thank the anonymous reviewer for helpful comments and valuable suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Zhongxi Zhang & Fu-Gui Shi

  2. School of Computer and Control Engineering, Yantai University, Yantai, Shandong, 264005, China

    Zhongxi Zhang

  3. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, China

    Qingguo Li

Corresponding author

Correspondence to Zhongxi Zhang.

Additional information

Publisher's Note

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

Rights and permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Shi, FG. & Li, Q. Continuity and Directed Completion of Topological Spaces. Order 39, 407–420 (2022). https://doi.org/10.1007/s11083-021-09586-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI : https://doi.org/10.1007/s11083-021-09586-z

Keywords

  • Weak monotone convergence space
  • Continuity
  • Quasicontinuity
  • Meet continuity
  • Δ-space
  • Scott completion

kitedonser.blogspot.com

Source: https://link.springer.com/content/pdf/10.1007/s11083-021-09586-z.pdf

0 Response to "Continuity Completion of Preimage is Included in Completion of"

Post a Comment

Subscribe to: Post Comments (Atom)

Iklan Atas Artikel

Iklan Tengah Artikel 1

Iklan Tengah Artikel 2

Iklan Bawah Artikel