Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T16:35:07.899Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified approach to some non-Hausdorff topological properties

Published online by Cambridge University Press:  16 March 2021

Qingguo Li ,
Zhenzhu Yuan Open the ORCID record for Zhenzhu Yuan [Opens in a new window]  and
Dongsheng Zhao
Qingguo Li*
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan410082, China
Zhenzhu Yuan
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan410082, China
Dongsheng Zhao
Affiliation:
Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, 637616, Singapore
*
*Corresponding author. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Sobriety, well-filteredness, and monotone convergence are three of the most important properties of topological spaces extensively studied in domain theory. Some other weak forms of sobriety and well-filteredness have also been investigated by some authors. In this paper, we introduce the notion of Θ-fine spaces, which provides a unified approach to such properties. In addition, this general approach leads to the definitions of some new topological properties.

Keywords

Sober spaced-spaceΘ-fine spacePF-well-filtered space
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 9 , October 2020 , pp. 997 - 1010
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S. (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. and Keimel, K. (2013). Quasicontinuous domains and the Smyth powerdomain. Electronic Notes in Theoretical Computer Science 298 215–232.CrossRefGoogle Scholar
Isbell, J. R. (1982). Completion of a construction of Johnstone. Proceedings of the American Mathematical Society 85 333–334.CrossRefGoogle Scholar
Jia, X. (2018). Meet-Continuity and Locally Compact Sober Dcpos. Phd thesis, University of Birmingham.Google Scholar
Jia, X., Jung, A. and Li, Q. (2016). A note on coherence of dcpos. Topology and its Applications 209 235–238.CrossRefGoogle Scholar
Johnstone, P. (1981). Scott is not always sober. In: Banaschewski, B. and Hoffmann, R. E. (eds.) Continuous Lattices, Proceedings Bremen 1979, Lecture Notes in Mathematics, vol. 871, Berlin, Springer-Verlag, 282–283.Google Scholar
Kou, H. (2001). U k-admitting dcpo’s need not be sober. In: Keimel, K., Zhang, G. Q., Liu, Y. M. and Chen, Y. X. (eds.) Domains and Processes, Semantic Structure on Domain Theory, vol. 1, Dordrecht, Kluwer Academic Publishers, 41–50.Google Scholar
Lu, C. and Li, Q. (2017). Weak well-filtered spaces and coherence. Topology and its Applications 230 373–380.CrossRefGoogle Scholar
Lu, C., Liu, D. and Li, Q. (2019). Equality of the Isbell and Scott topologies on function spaces of c-spaces. Houston Journal of Mathematics 45 265–284.Google Scholar
Schalk, A. (1993). Algebras for Generalized Power Constructions. Phd thesis, Technische Hochschule Darmstadt.Google Scholar
Xi, X. and Lawson, J. D. (2017). On well-filtered spaces and ordered sets. Topology and its Applications 228 139–144.CrossRefGoogle Scholar
Xu, X., Xi, X. and Zhao, D. (2021). A complete Heyting algebra whose Scott space is non-sober. Fundamenta Mathematicae 252 315–323.CrossRefGoogle Scholar
Zhao, D. and Fan, T. (2010). Dcpo-completion of posets. Theoretical Computer Science 411 (22) 2167–2173.CrossRefGoogle Scholar
Zhao, D. and Ho, W. K. (2015). On topologies defined by irreducible sets. Journal of Logical and Algebraic Methods in Programming 84 (1) 185–195.CrossRefGoogle Scholar