Kuratowski’s measure of non-compactness revisited. (English) Zbl 0672.54025

Lowen, R.

Q. J. Math., Oxf. II. Ser. 39, No. 154, 235-254 (1988).

The author earlier introduced the category of approach spaces with contractions, which contains the topological spaces with continuous maps as a bireflective and bicoreflective subcategory and the extended pseudo- quasi-metric spaces with non-expansive maps as a bicoreflective subcategory. A measure of non-compactness is defined for approach spaces and is shown to be effectively equivalent to the Kuratowski measure of non-compactness for extended pseudo-metric spaces. For X topological, \(m(X)=0\) if and only if X is compact. For X extended pseudo-quasi-metric, \(m(X)=0\) if and only if X is totally bounded. An arbitrary product (in the category of approach spaces) of totally bounded pseudo-quasi-metric spaces is totally bounded.

Reviewer: A.J.Klein

MSC:

54E50	Complete metric spaces
54B30	Categorical methods in general topology
54D30	Compactness

Keywords:

extended p-q metrics; total boundedness; approach spaces; extended pseudo-quasi-metric spaces; non-expansive maps; bicoreflective subcategory; measure of non-compactness