A set A in a metric space is called totally bounded if for each ε > 0 the set can be ε-approximated by a finite set. If this can be done, the finite set can always be chosen inside A. If the finite sets are replaced by an arbitrary approximating family of sets, this coincidence may disappear. We present necessary and sufficient conditions for the coincidence assuming only that the family is closed under finite unions. A complete analysis of the structure of totally bounded sets is presented in the case that the approximating family is a bornology, where approximation in either sense amounts to approximation in Hausdorff distance by members of the bornology. © 2008 Elsevier B.V. All rights reserved.
Beer, G., & Levi, S. (2009). Total boundedness and bornologies. TOPOLOGY AND ITS APPLICATIONS, 156(7), 1271-1288 [10.1016/j.topol.2008.12.030].
Total boundedness and bornologies
LEVI, SANDRO
2009
Abstract
A set A in a metric space is called totally bounded if for each ε > 0 the set can be ε-approximated by a finite set. If this can be done, the finite set can always be chosen inside A. If the finite sets are replaced by an arbitrary approximating family of sets, this coincidence may disappear. We present necessary and sufficient conditions for the coincidence assuming only that the family is closed under finite unions. A complete analysis of the structure of totally bounded sets is presented in the case that the approximating family is a bornology, where approximation in either sense amounts to approximation in Hausdorff distance by members of the bornology. © 2008 Elsevier B.V. All rights reserved.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
