Choice,hull, continuity and fidelity |
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| Affiliation: | 1. Lodz University of Technology, Faculty of Process and Environmental Engineering, Department of Environmental Engineering, 213 Wólczańska Str., 93–005 Łódź, Poland;2. TU Dortmund, Department of Biochemical and Chemical Engineering, Laboratory of Fluid Separations, Emil-Figge-Straße 70, 44227 Dortmund, Germany;3. TU Dortmund, Department of Biochemical and Chemical Engineering, Laboratory of Plant and Process Design, Emil-Figge-Straße 70, 44227 Dortmund, Germany;1. School of Medicine, University of Texas Medical Branch, Galveston, TX, USA;2. Department of Ophthalmology, Blanton Eye Institute, Houston Methodist Hospital, Houston, TX, USA;3. Departments of Ophthalmology, Neurology, and Neurosurgery, Weill Cornell Medicine, New York, NY, USA;4. The Neurology Center of Southern California, Carlsbad, CA, USA |
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| Abstract: | Continuity and fidelity (i.e., ‘path-independence’) conditions are studied for choices (picking subsets), hulls (picking supersets) and compositions of these. Examples of hulls are topological closure and convex hull, both of which are faithful. Using a continuity theorem of Sertel, a sufficient condition is given for closed convex hull, d, to be both continuous and faithful on the space of compact subsets of a locally convex topological vector space. A sufficient condition is also given for the joint continuity and fidelity of the composition, sd, of a choice, s, and d. In contrast with the Kalai and Megiddo theorem that singleton-valued maps of this form cannot be faithful and at the same time continuous on the space of finite subsets of En, the conjunction of (upper semi-)continuity and fidelity is shown to be commonplace for choices or maps of the above form (not constrained to be singleton-valued). |
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