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Keywords:
uniform space; approach uniform space; totally bounded; precompact; complete; measure of total boundedness; measure of completeness
uniform space; approach uniform space; totally bounded; precompact; complete; measure of total boundedness; measure of completeness
Summary:
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.
References:
[1] Čech E.: Topological Spaces. Interscience Publishers, 1966. MR 0211373
[2] Kuratowski C.: Sur les espaces complets. Fund. Math. 15 (1930), 301-309.
[3] Lowen R.: Kuratowski's measure of non-compactness revisited. Quarterly J. Math. Oxford 39 (1988), 235-254. MR 0947504 | Zbl 0672.54025
[4] Lowen R.: Approach spaces: a common supercategory of TOP and MET. Math. Nachrichten 141 (1989), 183-226. MR 1014427 | Zbl 0676.54012
[5] Lowen R.: Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, Oxford University Press, 1997. MR 1472024 | Zbl 0891.54001
[6] Lowen R., Windels B.: AUnif, a common supercategory of pMET and Unif. to appear in Int. J. Math. Math. Sci. MR 1486952 | Zbl 0890.54024
[7] Lowen R., Windels B.: Quantifying completion. submitted for publication. Zbl 0962.54023
[8] Preuss G.: Theory of Topological Structures. Kluwer Academic Publishers, 1987. MR 0937052 | Zbl 0649.54001
[9] Willard S.: General Topology. Addison Wesley Publishing Company, 1970. MR 0264581 | Zbl 1052.54001
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