Dedieu, Jean-Pierre L’image de la limite supérieure d’une famille d’ensembles est-elle égale à la limite supérieure de la famille des images? (Is the image of the upper limit of a family of sets equal to the upper limit of the family of images?). (French) Zbl 0729.47048 Ann. Fac. Sci. Toulouse, V. Sér., Math. 11, No. 2, 91-103 (1990). Summary: Let X, Y be Hausdorff topological vecto r spaces. Let I be a set of indices and let \({\mathcal F}\) be a filter in I. With each family \((A_ i)_{i\in I}\) of subsets of X is assocated its upper-limit: \[ \limsup_{{\mathcal F}}A_ i=\cap_{F\in {\mathcal F}}(\overline{\cup}_{i\in F}\overline{A_ i}). \] Let \((\Gamma_ j)_{j\in J}\) be a family of set-valued maps from X to Y and let \({\mathcal G}\) a filter in J. We give various conditions which ensure the inclusion: \[ \limsup_{{\mathcal F}\times {\mathcal G}}\Gamma_ j(A_ i)\subset (\limsup_{{\mathcal G}}\Gamma_ j)(\limsup_{{\mathcal F}}A_ i). \] Applications are given to optimization and mathematical morphology. Cited in 3 Documents MSC: 47H04 Set-valued operators 49J27 Existence theories for problems in abstract spaces 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) Keywords:upper-limit; family of set-valued maps; optimization; mathematical morphology × Cite Format Result Cite Review PDF Full Text: DOI Numdam Numdam EuDML References: [1] Bergé, C.) .- Espaces Topologiques et Fonctions Multivoques, Dunod. Paris (1966) · Zbl 0164.52902 [2] Dedieu, J.P.) . - Critères de fermeture pour l’image d’un fermé non convexe par une multiapplication, C. R. Acad. Sci., Paris, 287 (1978) pp. 941-943 · Zbl 0401.46005 [3] Dedieu, J.P.) .- Fermeture de l’image d’un fermé par une multiapplication, Multigraphié, Toulouse (1978) · Zbl 0401.46005 [4] Laurent, P.J.) .- Approximation et Optimisation, Hermann, Paris (1972) · Zbl 0238.90058 [5] Schmitt, M.) . - Quelques exemples d’analyse d’algorithmes de géométrie combinatoire par des techniques de morphologie mathématiques, Preprint (1988) (INRIA - BP 105 - 78153 Le Chesney Cedex (France)) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.