{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T07:09:08Z","timestamp":1756192148977},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T00:00:00Z","timestamp":1236124800000},"content-version":"unspecified","delay-in-days":4780,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[1996,2]]},"abstract":"<jats:p>This paper considers two main aspects of the lower power locale <jats:italic>P<jats:sub>L<\/jats:sub>(X)<\/jats:italic>: first, its relation to the symmetric topos construction of Bunge and Carboni; and second, its points, which, it is shown, are equivalent to the weakly closed sublocales of <jats:italic>X<\/jats:italic> with open domain. This is done as part of a more general discussion of arbitrary weakly closed sublocales, including a new characterization using suplattice homomorphisms from <jats:italic>o(X)<\/jats:italic> to <jats:italic>Sub(1)<\/jats:italic>, and a new proof of a theorem of Jibladze relating them to \u03a9-nuclei.<\/jats:p>","DOI":"10.1017\/s0960129500000876","type":"journal-article","created":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T04:00:07Z","timestamp":1236139207000},"page":"69-83","source":"Crossref","is-referenced-by-count":10,"title":["Constructive theory of the lower power locale"],"prefix":"10.1017","volume":"6","author":[{"given":"M.","family":"Bunge","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"funk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,3,4]]},"reference":[{"key":"S0960129500000876_ref024","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525902.017"},{"key":"S0960129500000876_ref023","volume-title":"Topology via Logic","author":"Vickers","year":"1989"},{"key":"S0960129500000876_ref022","unstructured":"Squire R. (1989) Presheaf Toposes and Propositional Logic, Ph. D. Thesis, McMaster University. Vermeulen, J. J. C. (1989) Weak Compactness in Constructive Spaces, Technical Report RR079, Department of Mathematics, University of Cape Town."},{"key":"S0960129500000876_ref025","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700012209"},{"key":"S0960129500000876_ref013","article-title":"An extension of the Galois theory of Grothendieck","volume":"309","author":"Joyal","year":"1984","journal-title":"Memoirs Amer. Math. Soc"},{"key":"S0960129500000876_ref008","first-page":"3","article-title":"Factorization theorems for localic geometric morphisms, I","volume":"22","author":"Johnstone","year":"1981","journal-title":"Cahiers de Top. G\u00e9om Diff. Cat\u00e9goriques"},{"key":"S0960129500000876_ref002","doi-asserted-by":"crossref","unstructured":"Bunge M. and Carboni A. (1994) The Symmetric Topos. To appear in Journal of Pure and Applied Algebra.","DOI":"10.1016\/0022-4049(94)00157-X"},{"key":"S0960129500000876_ref004","doi-asserted-by":"publisher","DOI":"10.1515\/9781400879915"},{"key":"S0960129500000876_ref005","first-page":"53","article-title":"The Display Locale of a cosheaf","volume":"36","author":"Funk","year":"1995","journal-title":"Cahiers de Top. G\u00e9om Diff. Cat\u00e9goriques"},{"key":"S0960129500000876_ref003","doi-asserted-by":"crossref","unstructured":"Bunge M. and Funk J. (1995) Spreads and the Symmetric Topos. To appear in Journal of Pure and Applied Algebra.","DOI":"10.1016\/0022-4049(94)00157-X"},{"key":"S0960129500000876_ref006","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-11409"},{"key":"S0960129500000876_ref011","first-page":"3","article-title":"A constructive \u2018closed subgroup theorem\u2019 for localic groups and groupoids","volume":"30","author":"Johnstone","year":"1989","journal-title":"Cahiers de Top. G\u00e9om. Diff. Cat\u00e9goriques"},{"key":"S0960129500000876_ref020","unstructured":"Robinson E. (1986) Power-Domains, Modalities and the Vietoris Monad, Technical Report No. 98, University of Cambridge Computer Laboratory."},{"key":"S0960129500000876_ref015","unstructured":"Kock A. (1994) Some Extensive Quantities in Locale Theory - a comment on a theorem of Bunge and Funk. Talk given at the Peripatetic Seminar on Sheaves and Logic, Aarhus, December 1994 (unpublished)."},{"key":"S0960129500000876_ref001","unstructured":"B\u00f6rger R. (1987) On the left adjoint from complete semi-lattices to frames, Haguen Preprint."},{"key":"S0960129500000876_ref007","first-page":"99","article-title":"The frame of fiberwise closed nuclei","volume":"32","author":"Jibladze","year":"1991","journal-title":"Cahiers de Top. G\u00e9om. Diff. Cat\u00e9goriques"},{"key":"S0960129500000876_ref009","volume-title":"Stone Spaces","author":"Johnstone","year":"1982"},{"key":"S0960129500000876_ref010","unstructured":"Johnstone P. T. (1985) Vietoris Locales and Localic Semilattices. In: Hoffman R. E. and Hoffman K. H. (eds.) Continuous Lattices and Their Applications, Marcel Dekker 155\u2013180."},{"key":"S0960129500000876_ref012","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525902.018"},{"key":"S0960129500000876_ref014","unstructured":"Kock A. (1993) Monads for which structures are adjoint to units. To appear in Journal of Pure and Applied Algebra."},{"key":"S0960129500000876_ref016","unstructured":"Lawvere F. W. (1983) Measures on Toposes. Lectures given at the Workshop on Category Theoretic Methods in Geometry, Aarhus, June 1983."},{"key":"S0960129500000876_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF02483860"},{"key":"S0960129500000876_ref018","first-page":"43","article-title":"On product and change of base for toposes","volume":"26","author":"Pitts","year":"1985","journal-title":"Cahiers de Top. G\u00e9om. Diff. 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