The lattice of smooth sublocales of a frame is isomorphic to the Bruns-Lakser completion of its locally closed sublocales.
Arrieta,A study of localic subspaces, separation, and variants of normality and their duals, PhD thesis, University of Coimbra and University of the Basque Country (UPV/EHU)
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The lattice of smooth sublocales as a Bruns-Lakser completion
The lattice of smooth sublocales of a frame is isomorphic to the Bruns-Lakser completion of its locally closed sublocales.