Well-posedness, bornologies, and the structure of metric spaces
Keywords:Well-posed problem, Bornology, UC-space, Cofinally complete space, Strong uniform continuity, Bornological convergence, Shielded from closed sets
Given a continuous nonnegative functional λ that makes sense defined on an arbitrary metric space (X, d), one may consider those spaces in which each sequence (xn) for which lim nâ†’âˆžλ(xn) = 0 clusters. The compact metric spaces, the complete metric spaces, the cofinally complete metric spaces, and the UC-spaces all arise in this way. Starting with a general continuous nonnegative functional λ defined on (X, d), we study the bornology Bλ of all subsets A of X on which limnâ†’âˆžλ(an) = 0 â‡’ (an) clusters, treating the possibility X âˆˆ Bλ as a special case. We characterize those bornologies that can be expressed as Bλ for some λ, as well as those that can be so induced by a uniformly continuous λ.
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