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

Doitchinov, Doitchin

Topology Appl. 30, No. 2, 127-148 (1988).

A quasi-metric on a set X is a non-negative real-valued function d defined on \(X\times X\) for which \(d(x,y)=0\) if and only if \(x=y\) and \(d(x,y)\leq d(x,y)+d(y,z)\) 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:

54E15	Uniform structures and generalizations
54E52	Baire category, Baire spaces
54C20	Extension of maps

Keywords:

Cauchy sequence; completion; completeness

References:

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[4]	Sieber, J. L.; Pervin, W. J., Completeness in quasi-uniform spaces, Math. Ann., 158, 79-81 (1965) · Zbl 0134.41702