Total boundedness and bornologies

被引：38

作者：

Beer, Gerald ^{[1
]}

Levi, Sandro ^{[2
]}

机构：

[1] Calif State Univ Los Angeles, Dept Math, Los Angeles, CA 90032 USA

[2] Univ Milano Bicocca, Dipartimento Matemat & Applicaz, I-20125 Milan, Italy

来源：

TOPOLOGY AND ITS APPLICATIONS | 2009年 / 156卷 / 07期

关键词：

Totally bounded set; Weakly totally bounded set; Bornology; Approximation in Hausdorff distance; CONVERGENCES; TOPOLOGIES;

D O I：

10.1016/j.topol.2008.12.030

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

A set A in a metric space is called totally bounded if for each epsilon > 0 the set call be epsilon-approximated by a finite set. If this call be done, the finite set call always be chosen inside A. If the finite sets are replaced by all 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. (C) 2008 Elsevier B.V. All rights reserved.

引用

页码：1271 / 1288

页数：18

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