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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 \(X\times X\) 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.

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 |

### References:

[1] | Fletcher, P.; Lindgren, W. F., Quasi-Uniform Spaces (1982), Marcel Dekker: Marcel Dekker New York and Basel · Zbl 0402.54024 |

[2] | Murdeshwar, M. G.; Naimpally, S. A., Quasi-Uniform Topological Spaces (1966), Noordhoff: Noordhoff Groningen · Zbl 0139.40501 |

[3] | Reilly, I. L.; Subrahmanyam, P. V.; Vamanamurthy, M. K., Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. Math., 93, 127-140 (1982) · Zbl 0472.54018 |

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

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