Abstract
The purpose of this paper is to identify the role of perfectness in the Michael insertion theorem for perfectly normal locales. We attain it by characterizing perfect locales in terms of strict insertion of two comparable lower semicontinuous and upper semicontinuous localic real functions. That characterization, when combined with the insertion theorem for normal locales, provides an improved formulation of the aforementioned pointfree form of Michael’s insertion theorem.
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Presented by W. Wm. McGovern.
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The authors acknowledge financial support from the Ministry of Economy and Competitiveness of Spain (Grant MTM2015-63608-P (MINECO/FEDER, UE)), from the Basque Government (Grant IT974-16) and from the Centre for Mathematics of the University of Coimbra (Grant UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and ERDF Partnership Agreement PT2020).
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Gutiérrez García, J., Kubiak, T. & Picado, J. Perfect locales and localic real functions. Algebra Univers. 81, 32 (2020). https://doi.org/10.1007/s00012-020-00661-x
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DOI: https://doi.org/10.1007/s00012-020-00661-x
Keywords
- Locale
- Sublocale
- Perfectness
- \(G_{\delta }\)-perfectness
- Perfect normality
- Semicontinuous real function
- Insertion theorem