Geodesic
Metric Bornologies and Kuratowski-Painleve Convergence to the Empty Set
Journal of convex analysis, Tome 8 (2001) no. 1, pp. 279-289.

Voir la notice de l'article provenant de la source Heldermann Verlag

Given a sequence {Tn} of nonempty closed sets Kuratowski-Painlevé convergent to the empty set in a noncompact metrizable space X, we show not only that there exists an admissible unbounded metric such that {Tn} converges to infinity in distance, but also that there must exist another such metric for which this is not the case. For such a sequence, let A consist of all subsets A of X whose closure hits Tn for at most finitely many indices n. We give necessary and sufficient conditions for A to be the family of bounded sets induced by some admissible metric for X, and show that all possible nontrivial metric bornologies for X arise in this manner if and only if the derived set of X is compact.
Classification : 54E35, 54B20
Mots-clés : Bounded set, metric bornology, Kuratowski-Painleve convergence, UC space, metric mode of convergence to infinity 
@article{JCA_2001_8_1_JCA_2001_8_1_a13,
     author = {G. Beer},
     title = {Metric {Bornologies} and {Kuratowski-Painleve} {Convergence} to the {Empty} {Set}},
     journal = {Journal of convex analysis},
     pages = {279--289},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {2001},
     url = {http://geodesic.mathdoc.fr/item/JCA_2001_8_1_JCA_2001_8_1_a13/}
}
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G. Beer. Metric Bornologies and Kuratowski-Painleve Convergence to the Empty Set. Journal of convex analysis, Tome 8 (2001) no. 1, pp. 279-289. http://geodesic.mathdoc.fr/item/JCA_2001_8_1_JCA_2001_8_1_a13/