Well-posedness, bornologies, and the structure of metric spaces

Gerald Beer, Manuel Segura


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 n→∞λ(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 limn→∞λ(an) = 0 ⇒ (an) clusters, treating the possibility X ∈ 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.


Well-posed problem; Bornology; UC-space; Cofinally complete space; Strong uniform continuity; Bornological convergence; Shielded from closed sets

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1. Some properties of bornological convergences
Jesús Rodríguez-López, M.A. Sánchez-Granero
Topology and its Applications  vol: 158  issue: 1  first page: 101  year: 2011  
doi: 10.1016/j.topol.2010.10.009

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

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