Well-posedness, bornologies, and the structure of metric spaces
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 th...
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Universitat Politècnica de València
2009-04-01
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| Series: | Applied General Topology |
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| Online Access: | http://polipapers.upv.es/index.php/AGT/article/view/1793 |
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doaj-55b95e1e4a5d492a8d8a9b20f102b6952020-11-24T22:52:41ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472009-04-0110113115710.4995/agt.2009.17931451Well-posedness, bornologies, and the structure of metric spacesGerald Beer0Manuel Segura1California State University Los AngelesCalifornia State University Los AngelesGiven 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.http://polipapers.upv.es/index.php/AGT/article/view/1793Well-posed problemBornologyUC-spaceCofinally complete spaceStrong uniform continuityBornological convergenceShielded from closed sets |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Gerald Beer Manuel Segura |
| spellingShingle |
Gerald Beer Manuel Segura Well-posedness, bornologies, and the structure of metric spaces Applied General Topology Well-posed problem Bornology UC-space Cofinally complete space Strong uniform continuity Bornological convergence Shielded from closed sets |
| author_facet |
Gerald Beer Manuel Segura |
| author_sort |
Gerald Beer |
| title |
Well-posedness, bornologies, and the structure of metric spaces |
| title_short |
Well-posedness, bornologies, and the structure of metric spaces |
| title_full |
Well-posedness, bornologies, and the structure of metric spaces |
| title_fullStr |
Well-posedness, bornologies, and the structure of metric spaces |
| title_full_unstemmed |
Well-posedness, bornologies, and the structure of metric spaces |
| title_sort |
well-posedness, bornologies, and the structure of metric spaces |
| publisher |
Universitat Politècnica de València |
| series |
Applied General Topology |
| issn |
1576-9402 1989-4147 |
| publishDate |
2009-04-01 |
| description |
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. |
| topic |
Well-posed problem Bornology UC-space Cofinally complete space Strong uniform continuity Bornological convergence Shielded from closed sets |
| url |
http://polipapers.upv.es/index.php/AGT/article/view/1793 |
| work_keys_str_mv |
AT geraldbeer wellposednessbornologiesandthestructureofmetricspaces AT manuelsegura wellposednessbornologiesandthestructureofmetricspaces |
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1725664997408768000 |