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On completeness in quasi-metric spaces. (English) Zbl 0668.54019

A quasi-metric on a set X is a non-negative real-valued function d defined on X×X for which d(x,y)=0 if and only if x=y and d(x,y)d(x,y)+d(y,z) for any x, y, and z in X. Each metric on X is clearly a quasi-metric, and each quasi-metric on X induces a quasi- uniformity and quasi-uniform topology in the usual manner.

The author provides a thorough discussion of the motivation for and limitations of the notions of Cauchy sequence and completion for quasi- metric spaces. After presenting suitable definitions of Cauchy sequence and completeness, he is able to obtain a satisfactory theory for a standard completion of quasi-metric spaces, but only in the specialized category of balanced quasi-metric spaces. The appropriate expected theorems, including those which guarantee unique extension of quasi- uniformly continuous mappings to the completions, reinforce the naturality of the demonstrated completion process.

Reviewer: S.C.Carlson

MSC:
54E15Uniform structures and generalizations
54E52Baire category, Baire spaces
54C20Extension of maps on topological spaces