Abstract
In this paper, we focus on p-sober spaces and prove that (1) the T0 space X is p-sober if and only if the Smyth power space of X is p-sober; (2) the space X has a p-sober dcpo model if and only if X is T1 and p-sober; (3) every non-p-sober T0 space does not have a p-sobrification; (4) the T0 space X is sober if and only if X is p-sober and PD.
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References
Adámek, J., et al.: Abstract and Concrete Categories: The Joy of Cats, Wiley, New York, 1990
Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order, Second Edition, Cambridge University Press, New York, 2002
Dowker, C. H., Papert, D.: Quotient frames and subspaces. Proc. London Math. Soc., 3(16), 275–296 (1966)
Engelking, R.: General Topology, Polish Scientific Publishers, Warszawa, 1977
Gierz, G., et al.: Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press, Cambridge, 2003
Goubault-Larrecq, J.: Non-Hausdorff Topology and Domain Theory, Cambridge University Press, Cambrige, 2013
Heckmann, R., Keimel, K.: Quasicontinuous domains and the Smyth powerdomain. Electron. Notes Theor. Comput. Sci., 298, 215–232 (2013)
Ho, W. K., et al.: The Ho–Zhao problem. Log. Methods Comput. Sci., 14, 1–19 (2018)
Isbell, J.: Completion of a construction of Johnstone. Proc. Amer. Math. Soc., 85, 333–334 (1982)
Johnstone, P. T.: Scott is not always sober, In: Continuous Lattices. Lecture Notes in Mathematics, Springer-Verlag, Berlin, 282–283 (1981)
Johnstone, P. T.: Stone Spaces, Cambridge University Press, Cambridge, 1982
Jia, X. D.: Meet-continuity and locally compact sober dcpos, Ph.D. Thesis, University of Birmingham, 2018
Keimel, K., Lawson, J. D.: D-completions and the d-topology. Ann. Pure Appl. Logic., 159, 292–306 (2009)
Lu, J., et al.: Nonexistence of fc-bounded sobrification. arXiv:2011.11606 (2020)
Nel, L., Wilson, R.: Epireflections in the category of T0-spaces. Fund. Math., 75(1), 69–74 (1972)
Schalk, A.: Algebras for Generalized Power Constructions, Ph.D. Thesis, Technische Hochschule Darmstadt, 1993
Shen, C., et al.: The reflectivity of some categories of To spaces in domain theory. arXiv:2110.01138 (2021)
Shen, C., et al.: The non-reflectivity of open well-filtered spaces via 6-topology. Houston J. Math., 48(4), 843–854 (2022)
Skula, L.: On a reflective subcategory of the category of all topological Spaces. Trans. Amer. Math. Soc., 142, 37–41 (1969)
Xi, X. Y., Lawson, J.: On well-filtered spaces and ordered sets. Topology Appl., 228, 139–144 (2017)
Xu, X. Q., Zhao, D. S.: On topological Rudin’s lemma, well-filtered spaces and sober spaces. Topology Appl., 272, 107080 (2020)
Zhao, D. S.: Poset models of topological spaces, In: Proceeding of International Conference on Quantitative Logic and Quantification of Software, Global-Link Publisher, Hong Kong, 229–238 (2009)
Zhao, D. S., Fan, T. H.: Dcpo-completion of posets. Theoret. Comput. Sci., 411, 2167–2173 (2010)
Zhao, D.S., Ho, W.K.: On topologies defined by irreducible sets. J. Log. Algebr. Program., 84, 185–195 (2015)
Zhao, D. S., Xi, X. Y.: Directed complete poset models of T1 spaces. Math. Proc. of the Cambridge Philos. Soc., 164(1), 125–134 (2018)
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The authors would like to thank the editor and two anonymous reviewers for their valuable comments and suggestions.
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Supported by the National Natural Science Foundation of China (Grant No. 11531009) 1) Corresponding author
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Zhang, Y.J., Zhao, B. On P-sober Spaces. Acta. Math. Sin.-English Ser. 39, 1768–1780 (2023). https://doi.org/10.1007/s10114-023-2197-4
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DOI: https://doi.org/10.1007/s10114-023-2197-4