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New relation defines F λ -perfect and F λ -regular locales, highlighting their structure and connections.
We introduce a new relation on locales, the F λ -below relation < λ , obtained from F λ -sublocales. This relation yields intrinsic definitions of F λ -perfect and F λ -regular locales, whose structural properties and interrelations are investigated. We further provide two constructions of F λ -regular locales, one of which establishes that the category of F λ -regular locales with C ∞ -maps forms a reflective subcategory of the category of locales with < λ -maps.
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Mbekezeli Nxumalo (2026) studied this question.
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