β-Normality in locales
Submitted: 2024-11-12
|Accepted: 2025-02-24
|Published: 2025-04-01
Copyright (c) 2025 Mbekezeli Nxumalo, Thobile Ngcamphalala
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
β-normal, α-normal, almost weakly β-normal, locale, normal
Supporting agencies:
National Research foundation of South Africa
Abstract:
In this paper, we establish the theory of β-normal locales which are the point- free counterparts of β-normal spaces which were introduced by Arkhangel’skii and Ludwig. We give characterizations of β-normal locales using some types of open sublocales. Certain circumstances are exhibited in which normality coincides with β-normality. For instance, we use the localic Kateˇtov-Tong insertion theorem to prove that every β-normal locale which is also a coframe is normal. This result argues that there does not exist a finite β-normal locale which is not normal. Included here is also an answer to Murtinova ́’s question about the existence of a regular β-normal space which is not Tychonoff. The study of β-normal locales leads to some variants of β-normal locales, namely α-normal locales and almost weakly β-normal locales. This paper also examines preservation and reflection of β-normal locales by localic maps.
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