Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A --> d(x,A) is continuous for each x is-an-element-of X and each admissible metric d, We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) --> --> inf{d(a, b): a is-an-element-of A, b is-an-element-of B} is continuous on CL(X) x CL(X)

Beer, G., Lechicki, A., Levi, S., Naimpally, S. (1992). Distance functionals and suprema of hyperspace topologies. ANNALI DI MATEMATICA PURA ED APPLICATA, 162, 367-381 [10.1007/BF01760016].

Distance functionals and suprema of hyperspace topologies

Levi, S;
1992

Abstract

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A --> d(x,A) is continuous for each x is-an-element-of X and each admissible metric d, We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) --> --> inf{d(a, b): a is-an-element-of A, b is-an-element-of B} is continuous on CL(X) x CL(X)
Articolo in rivista - Articolo scientifico
Vietoris topology; weak topology for uniformly equivalent metrics
English
1992
162
367
381
none
Beer, G., Lechicki, A., Levi, S., Naimpally, S. (1992). Distance functionals and suprema of hyperspace topologies. ANNALI DI MATEMATICA PURA ED APPLICATA, 162, 367-381 [10.1007/BF01760016].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/18634
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