Let S be an ideal of subsets of a metric space (X, d). A net of subsets (Aλ) of X is called S-convergent to a subset A of X if for each S ∈ S and each ε &gt; 0, we have eventually A ∩ S ⊆ A <sub>λ</sub><sup>ε</sup> and Aλ ∩ S ⊆. A <sup>ε</sup>. We identify necessary and sufficient conditions for this convergence to be admissible and topological on the power set of X. We show that S-convergence is compatible with a pseudometrizable topology if and only if S has a countable base and each member of S has an ε-enlargement that is again in S. Further, in the case that the ideal is a bornology, we show that S-convergence when pseudometrizable is Attouch-Wets convergence with respect to an equivalent metric.

Beer, G., Levi, S. (2008). Pseudometrizable bornological convergence is Attouch-Wets convergence. JOURNAL OF CONVEX ANALYSIS, 15(2), 439-453.

Pseudometrizable bornological convergence is Attouch-Wets convergence

LEVI, SANDRO
2008

Abstract

Let S be an ideal of subsets of a metric space (X, d). A net of subsets (Aλ) of X is called S-convergent to a subset A of X if for each S ∈ S and each ε > 0, we have eventually A ∩ S ⊆ A λε and Aλ ∩ S ⊆. A ε. We identify necessary and sufficient conditions for this convergence to be admissible and topological on the power set of X. We show that S-convergence is compatible with a pseudometrizable topology if and only if S has a countable base and each member of S has an ε-enlargement that is again in S. Further, in the case that the ideal is a bornology, we show that S-convergence when pseudometrizable is Attouch-Wets convergence with respect to an equivalent metric.
Articolo in rivista - Articolo scientifico
Bornological convergence, Attouch-Wets convergence, hyperspace, bornology
English
2008
15
2
439
453
none
Beer, G., Levi, S. (2008). Pseudometrizable bornological convergence is Attouch-Wets convergence. JOURNAL OF CONVEX ANALYSIS, 15(2), 439-453.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/5788
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