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

Beer, Gerald ^[1] ; Segura, Manuel ^[1]
1. [1] California State University Los Angeles
  
  California State University Los Angeles
  
  Estados Unidos
Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 10, Nº. 1, 2009, págs. 131-157
Idioma: inglés
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Resumen
- 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.