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Hofmann-Mislove type definitions of non-Hausdorff spaces

Published online by Cambridge University Press:  27 June 2022

Chong Shen*
School of Science, Beijing University of Posts and Telecommunications, Beijing, China
Xiaoyong Xi
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu, Yancheng, China
Xiaoquan Xu
School of Mathematics and Statistics, Minnan Normal University, Fujian, Zhangzhou, China
Dongsheng Zhao
Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore
*Corresponding author. Email:


One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $\Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $\Psi$ -well-filtered spaces is provided via $\Psi$ -sets. We also discuss the relationship between $\Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $\Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $\Psi$ that has a designated property, we show that a $T_0$ space X is $\Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $\Psi$ -well-filtered.

© The Author(s), 2022. Published by Cambridge University Press

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This research was supported by the National Natural Science Foundation of China (Nos. 1210010153, 12071199, 12071188, 11661057, 11871097), Jiangsu Provincial Department of Education (21KJB110008) and the Natural Science Foundation of Jiangxi Province, China (No. 20192ACBL20045).


Engelking, R. (1989). General Topology , Sigma Series in Pure Mathematics, vol. 6, Berlin, Heldermann Verlag.Google Scholar
Erné, M. (2009). Infinite distributive laws versus local connectedness and compactness properties. Topology and its Applications 156 20542069.CrossRefGoogle Scholar
Eršhov, Yu. L. (1999). On d-spaces. Theoretical Computer Science 224 5972.Google Scholar
Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M. and Scott, D. (2003). Continuous lattices and Domains , Encyclopedia of Mathematics and Its Applications, vol. 93, Cambridge, Cambridge University Press.Google Scholar
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Heckmann, R. (1991). An upper power domain construction in terms of strongly compact sets. In: International Conference on Mathematical Foundations of Programming Semantics, Berlin, Heidelberg, Springer, 272293.Google Scholar
Heckmann, R. and Keimel, K. (2013). Quasicontinuous domains and the Smyth powerdomain. Electronic Notes in Theoretical Computer Science 298 215232.CrossRefGoogle Scholar
Hoffmann, R. E. (1981). Projective sober spaces. In: Banaschewski, B. and Hoffmann, R. E. (eds.) Continuous Lattices, Lecture Notes in Mathematics, vol. 871, Berlin, Heidelberg, Springer, 105 125158.Google Scholar
Hoffmann, R. (1979). On the sobrification remainder $X^s-X$ . Pacific Journal of Mathematics 83 145156.CrossRefGoogle Scholar
Jia, X. and Jung, A. (2016). A note on coherence of dcpos. Topology and its Applications 209 235238.CrossRefGoogle Scholar
Keimel, K. and Lawson, J. (2009) D-topology and d-completion. Annals of Pure and Applied Logic 159 292306.CrossRefGoogle Scholar
Li, Q., Yuan, Z. and Zhao, D. (2021). A unified approach to some non-Hausdorff topological properties. Mathematical Structures in Computer Science 30 (9) 9971010.CrossRefGoogle Scholar
Liu, B., Li, Q. and Wu, G. (2020). Well-filterifications of topological spaces. Topology and its Applications 279 107245.Google Scholar
Schalk, A. (1993). Algebras for Generalized Power Constructions. Phd thesis, Technische Hochschule Darmstadt.Google 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.CrossRefGoogle Scholar
Shen, C., Xi, X., Xu, X. and Zhao, D. (2020). On open well-filtered spaces. Logical Methods in Computer Science 16 (4) 418.Google Scholar
Wu, G., Xi, X., Xu, X. and Zhao, D. (2020). Existence of well-filtered reflections of $T_0$ topological spaces. Topology and its Applications 270 107044.CrossRefGoogle Scholar
Wyler, O. (1981). Dedekind complete posets and Scott topologies. In: Continuous Lattices, Berlin, Heidelberg, Springer, 384389.CrossRefGoogle Scholar
Xu, X. (2021). On H-sober spaces and H-sobrifications of $T_0$ spaces. Topology and its Applications 289 107548.Google Scholar
Xu, X., Shen, C., Xi, X. and Zhao, D. (2020a). On $T_0$ spaces determined by well-filtered spaces. Topology and its Applications 282 107323.CrossRefGoogle Scholar
Xu, X., Shen, C., Xi, X. and Zhao, D. (2020b). First countability, -well-filtered spaces and reflections. Topology and its Applications 279 107255.CrossRefGoogle Scholar
Xu, X., Xi, X. and Zhao, D. (2021). A complete Heyting algebra whose Scott space is non-sober. Fundamenta Mathematicae 252 315323.Google Scholar