On the quantification of uniform properties

Robert Lowen; Bart Windels

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 4, page 749-759
  • ISSN: 0010-2628

Abstract

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Approach spaces ([4], [5]) turned out to be a natural setting for the quantification of topological properties. Thus a measure of compactness for approach spaces generalizing the well-known Kuratowski measure of non-compactness for metric spaces was defined ([3]). This article shows that approach uniformities (introduced in [6]) have the same advantage with respect to uniform concepts: they allow a nice quantification of uniform properties, such as total boundedness and completeness.

How to cite

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Lowen, Robert, and Windels, Bart. "On the quantification of uniform properties." Commentationes Mathematicae Universitatis Carolinae 38.4 (1997): 749-759. <http://eudml.org/doc/22304>.

@article{Lowen1997,
abstract = {Approach spaces ([4], [5]) turned out to be a natural setting for the quantification of topological properties. Thus a measure of compactness for approach spaces generalizing the well-known Kuratowski measure of non-compactness for metric spaces was defined ([3]). This article shows that approach uniformities (introduced in [6]) have the same advantage with respect to uniform concepts: they allow a nice quantification of uniform properties, such as total boundedness and completeness.},
author = {Lowen, Robert, Windels, Bart},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {uniform space; approach uniform space; totally bounded; precompact; complete; measure of total boundedness; measure of completeness; Cauchy ideal},
language = {eng},
number = {4},
pages = {749-759},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the quantification of uniform properties},
url = {http://eudml.org/doc/22304},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Lowen, Robert
AU - Windels, Bart
TI - On the quantification of uniform properties
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 4
SP - 749
EP - 759
AB - Approach spaces ([4], [5]) turned out to be a natural setting for the quantification of topological properties. Thus a measure of compactness for approach spaces generalizing the well-known Kuratowski measure of non-compactness for metric spaces was defined ([3]). This article shows that approach uniformities (introduced in [6]) have the same advantage with respect to uniform concepts: they allow a nice quantification of uniform properties, such as total boundedness and completeness.
LA - eng
KW - uniform space; approach uniform space; totally bounded; precompact; complete; measure of total boundedness; measure of completeness; Cauchy ideal
UR - http://eudml.org/doc/22304
ER -

References

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  1. Čech E., Topological Spaces, Interscience Publishers, 1966. Zbl0141.39401MR0211373
  2. Kuratowski C., Sur les espaces complets, Fund. Math. 15 (1930), 301-309. (1930) Zbl56.1124.04
  3. Lowen R., Kuratowski's measure of non-compactness revisited, Quarterly J. Math. Oxford 39 (1988), 235-254. (1988) Zbl0672.54025MR0947504
  4. Lowen R., Approach spaces: a common supercategory of TOP and MET, Math. Nachrichten 141 (1989), 183-226. (1989) Zbl0676.54012MR1014427
  5. Lowen R., Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997. Zbl0891.54001MR1472024
  6. Lowen R., Windels B., AUnif, a common supercategory of pMET and Unif, to appear in Int. J. Math. Math. Sci. Zbl0890.54024MR1486952
  7. Lowen R., Windels B., Quantifying completion, submitted for publication. Zbl0962.54023
  8. Preuss G., Theory of Topological Structures, Kluwer Academic Publishers, 1987. Zbl0649.54001MR0937052
  9. Willard S., General Topology, Addison Wesley Publishing Company, 1970. Zbl1052.54001MR0264581

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