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Characterizations of $\omega$-Rudin spaces via sequence convergence

Published online by Cambridge University Press:  09 December 2025

Xiaoquan Xu Open the ORCID record for Xiaoquan Xu [Opens in a new window]
Xiaoquan Xu*
Affiliation:
Fujian Key Laboratory of Granular Computing and Applications, Minnan Normal University, Zhangzhou, 363000, China
*
Email: xiqxu2002@163.com
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Abstract

The author’s primary goal in this paper is to characterize $\omega$-Rudin sets and $\omega$-Rudin spaces via sequence convergence and give some important applications of such characterizations. For an irreducible closed set $A$ of a $T_0$-space $X$, we prove that the following four conditions are equivalent: (1) $A$ is an $\omega$-Rudin set; (2) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(a_n)_{n\in \mathbb{N}}$ simultaneously converges to all points of $A$; (3) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the Hoare power space of $X$; (4) there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the sobrification of $X$. Based on these characterizations, we obtain some characterizations of $\omega$-Rudin spaces and sober spaces. In particular, we show that for a complete lattice $L$, its Scott space $\Sigma L$ is sober iff for any nonempty Scott irreducible closed set $A$ of $L$, there is $\{a_n : n\in \mathbb{N}\}\subseteq A$ such that the sequence $(a_n)_{n\in \mathbb{N}}$ simultaneously Scott converges to all points of $A$ or, equivalently, the sequence $(\overline {\{a_n\}})_{n\in \mathbb{N}}$ converges to $A$ in the sobrification of $\Sigma L$. Several related examples are presented. We also investigate some basic properties of $\omega$-Rudin spaces. It is proved that the property of being an $\omega$-Rudin space is retractive, productive, and closed-hereditary. We give two examples to show that it is not saturated-hereditary and the category $\boldsymbol{\omega }$-$\mathbf{Rud}$ of $\omega$-Rudin spaces does not have equalizers, and hence, $\boldsymbol{\omega }$-$\mathbf{Rud}$ is not reflective in the category $\mathbf{Top}_{0}$ of all $T_0$-spaces. Finally, we discuss the Smyth power spaces of $\omega$-Rudin spaces. It is shown that if the Smyth power space of a $T_0$-space $X$ is an $\omega$-Rudin space, then $X$ is an $\omega$-Rudin space. The question naturally arises whether the Smyth power space of an $\omega$-Rudin space is still an $\omega$-Rudin space.

Keywords

Sequence convergenceω-Rudin setω-Rudin spacesobrietyScott topologyHoare power spaceSmyth power space

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Type
Paper
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Mathematical Structures in Computer Science , Volume 35 , 2025 , e37
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© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

†

This research was supported by the National Natural Science Foundation of China (Nos. 12471070, 12071199).

References

Abramsky, S. and Jung, A. (1994). Domain theory. In: Semantic Structures, Handbook of Logic in Computer Science, vol. 3, Warzawa, Polish Scientific Publishers, 1–168.Google Scholar
Artin, M., Grothendieck, A. and Verdier, J. (1972). Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, vol. 269, Berlin-Heidelberg-New York, Springer.Google Scholar
Blanksma, T. (1968). Lattice characterizations and compactifications. Doctoral dissertation, Rijksuniversiteit te Utrecht.Google Scholar
Brecht, M. and Kawai, T. (2019). On the commutativity of the powerspace constructions. Logical Methods in Computer Science 15 (3) 25.Google Scholar
Engelking, R. (1989). General Topology, Warzawa, Polish Scientific Publishers.Google Scholar
Erné, M. (2018). Categories of locally hypercompact spaces and quasicontinuous posets. Applied Categorical Structures 26 823–854.10.1007/s10485-018-9536-0CrossRefGoogle Scholar
Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M. and Scott, D. (2003). Continuous Lattices and Domains, Cambridge, Cambridge University Press.10.1017/CBO9780511542725CrossRefGoogle Scholar
Gierz, G., Lawson, J. and Stralka, A. (1983). Quasicontinuous posets. Houston Journal of Mathematics 9 191–208.Google Scholar
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory, New Mathematical Monographs, vol. 22, Cambridge, Cambridge University Press.10.1017/CBO9781139524438CrossRefGoogle Scholar
Heckmann, R. (1990). Power domain constructions. PhD thesis, Universität des Saarlandes.Google Scholar
Heckmann, R. (1992). An upper power domain construction in terms of strongly compact sets. In: Mathematical Foundations of Programming Semantics, Lecture Notes in Computer Science, vol. 598, Berlin, Springer-Verlag, 272–293.10.1007/3-540-55511-0_14CrossRefGoogle Scholar
Heckmann, R. and Keimel, K. (2013). Quasicontinuous domains and the Smyth powerdomain. Electronic Notes in Theoretical Computer Science 298 215–232.10.1016/j.entcs.2013.09.015CrossRefGoogle Scholar
Hoffmann, R. (1979). Sobrification of partially ordered sets. Semigroup Forum 17 123–138.10.1007/BF02194315CrossRefGoogle Scholar
Hofmann, K. and Lawson, J. (1978). The spectral theory of distributive continuous lattices Transactions of the American Mathematical Society 246 285–310.10.1090/S0002-9947-1978-0515540-7CrossRefGoogle Scholar
Hou, H., Miao, H. and Li, Q. (2024). The order- K -ification monads. Mathematical Structures in Computer Science 34 45–62.10.1017/S0960129523000403CrossRefGoogle Scholar
Jia, X. (2018). Meet-Continuity and Locally Compact Sober Dcpos. PhD thesis, University of Birmingham.Google Scholar
Johnstone, P. (1981). Scott is not always sober. In: Continuous Lattices, Lecture Notes in Math, vol. 871, Berlin, Springer-Verlag, 282–283.10.1007/BFb0089911CrossRefGoogle Scholar
Keimel, K. and Lawson, J. (2009). $D$ -completions and the $d$ -topology. Annals of Pure and Applied Logic 159 (3) 292–306.10.1016/j.apal.2008.06.019CrossRefGoogle Scholar
Kou, H. (2001). Uk-admitting dcpos need not be sober. In: Domains and Processes, Semantic Structure On Domain Theory, vol. 1, Netherlands, Kluwer Academic Publishers, 41–50.10.1007/978-94-010-0654-5_3CrossRefGoogle Scholar
Lawson, J., Wu, G. and Xi, X. (2020). Well-filtered spaces, compactness, and the lower topology. Houston Journal of Mathematics 46 (1) 283–294.Google Scholar
Liu, B., Li, Q. and Wu, G. (2020). Well-filterifications of topological spaces. Topology and its Applications 279 107245.10.1016/j.topol.2020.107245CrossRefGoogle Scholar
Lu, C. and Li, Q. (2017). Weak well-filtered spaces and coherence. Topology and its Applications 230 373–380.10.1016/j.topol.2017.08.049CrossRefGoogle Scholar
MacLane, S. (1997). Categories for the Working Mathematician, 2nd edn. Springer, 1997.Google Scholar
Miao, H., Xi, X., Li, Q. and Zhao, D. (2023). Not every countable complete lattice is sober. Mathematical Structures in Computer Science 33 809–831.10.1017/S0960129523000269CrossRefGoogle Scholar
Miao, M., Li, Q. and Zhao, D. (2021). On two problems about sobriety of topological spaces. Topology and its Applications 295 107667.10.1016/j.topol.2021.107667CrossRefGoogle Scholar
Rudin, M. (1980). Directed sets which converge. In: General Topology and Modern Analysis, University of California, Riverside, Academic Press, 305–307, 1980.Google Scholar
Schalk, A. (1993). Algebras for generalized power constructions. PhD thesis, Technische Hochschule Darmstadt.Google Scholar
Shan, Q., Bao, M., Wen, X. and Xu, X. (2022). On almost sober spaces. Topology and its Applications 305 107896.10.1016/j.topol.2021.107896CrossRefGoogle Scholar
Shen, C., Xi, X., Xu, X. and Zhao, D. (2019). On well-filtered reflections of $T_0$ spaces. Topology and its Applications 267 106869.10.1016/j.topol.2019.106869CrossRefGoogle Scholar
Smyth, M. (1978). Power domains. Journal of Computer and System Sciences 16 23–36.10.1016/0022-0000(78)90048-XCrossRefGoogle Scholar
Wyler, U. (1981). Dedekind complete posets and Scott topologies. In: Continuous Lattices, Lecture Notes in Math, vol. 871, Berlin, Springer-Verlag, 384–389.10.1007/BFb0089920CrossRefGoogle Scholar
Xi, X. and Lawson, J. (2017). On well-filtered spaces and ordered sets. Topology and its Applications 228 139–144.10.1016/j.topol.2017.06.002CrossRefGoogle Scholar
Xu, X., Miao, H. and Li, Q. (2025). Fréchet spaces, $\omega$ -Rudin property and Smyth power spaces. Topology and its Applications 363 109235, 1–19.10.1016/j.topol.2025.109235CrossRefGoogle Scholar
Xu, X., Shen, C., Xi, X. and Zhao, D. (2020a). First countability, $\omega$ -well-filtered spaces and reflections. Topology and its Applications 279 107255.10.1016/j.topol.2020.107255CrossRefGoogle Scholar
Xu, X., Shen, C., Xi, X. and Zhao, D. (2020b). On $T_{0}$ spaces determined by well-filtered spaces. Topology and its Applications 282 107323.10.1016/j.topol.2020.107323CrossRefGoogle Scholar
Xu, X., Shen, C., Xi, X. and Zhao, D. (2021a). First-countability, $\omega$ -Rudin spaces and well-filtered determined. Topology and its Applications 300 107775.10.1016/j.topol.2021.107775CrossRefGoogle Scholar
Xu, X., Wen, X. and Xi, X. (2023). Scott topology on Smyth power posets. Mathematical Structures in Computer Science 33 832–867.10.1017/S0960129523000257CrossRefGoogle Scholar
Xu, X., Xi, X. and Zhao, D. (2021b). A complete Heyting algebra whose Scott topology is non-sober. Fundamenta Mathematicae 252 315–323.10.4064/fm704-4-2020CrossRefGoogle Scholar
Xu, X. and Zhao, D. (2021). Some open problems on well-filtered spaces and sober spaces. Topology and its Applications 301 1–19.10.1016/j.topol.2020.107540CrossRefGoogle Scholar
Zhang, Y. and Zhao, B. (2023). On $p$ -sober spaces. Acta Mathematica Sinica 39 (9) 1768–1780.10.1007/s10114-023-2197-4CrossRefGoogle Scholar
Zhao, D. and Ho, W. (2015). On topologies defined by irreducible sets. Journal of Logical and Algebraic Methods in Programming 84 (1) 185–195.10.1016/j.jlamp.2014.10.003CrossRefGoogle Scholar