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Hofmann-Mislove type definitions of non-Hausdorff spaces

Published online by Cambridge University Press:  27 June 2022

Chong Shen*
Affiliation:
School of Science, Beijing University of Posts and Telecommunications, Beijing, China
Xiaoyong Xi
Affiliation:
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu, Yancheng, China
Xiaoquan Xu
Affiliation:
School of Mathematics and Statistics, Minnan Normal University, Fujian, Zhangzhou, China
Dongsheng Zhao
Affiliation:
Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore
*
*Corresponding author. Email: shenchong0520@163.com

Abstract

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.

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

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Footnotes

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).

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