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Scott topology on Smyth power posets

Published online by Cambridge University Press:  27 July 2023

Xiaoquan Xu Open the ORCID record for Xiaoquan Xu [Opens in a new window] ,
Xinpeng Wen  and
Xiaoyong Xi
Xiaoquan Xu*
Affiliation:
Fujian Key Laboratory of Granular Computing and Applications, Minnan Normal University, Zhangzhou, China
Xinpeng Wen
Affiliation:
College of Mathematics and Information, Nanchang Hangkong University, Nanchang, China
Xiaoyong Xi
Affiliation:
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, China
*
Corresponding author: Xiaoquan Xu; Email: xiqxu2002@163.com
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Abstract

For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$. $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$. A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated.

Keywords

Scott topologySmyth power posetSmyth power spacesobrietywell-filterednesslocal compactnessfirst-countability
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 33 , Issue 9 , October 2023 , pp. 832 - 867
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

†

This research was supported by the National Natural Science Foundation of China (Nos. 12071199, 12071188, 11661057).

Dedicated to Professor Dana Scott on the occasion of his 90th birthday

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