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

• Main Authors: Gerald Beer, Manuel Segura
• Format: Article
• Language: English
• Published: Universitat Politècnica de València 2009-04-01
• Series: Applied General Topology
• Subjects: Well-posed problem; Bornology; UC-space; Cofinally complete space; Strong uniform continuity; Bornological convergence; Shielded from closed sets
• Online Access: http://polipapers.upv.es/index.php/AGT/article/view/1793
• ISSN: 1576-9402; 1989-4147

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.