Weighted quasi-metrics.

*(English)*Zbl 0915.54023
Andima, Susan (ed.) et al., Papers on general topology and applications. Papers from the 8th summer conference at Queens College, New York, NY, USA, June 18–20, 1992. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 728, 64-77 (1994).

Summary: The authors study the class of topologies which are induced by weighted quasi-metrics (equivalently, partial metrics). Partial metrics were introduced by S. G. Matthews in his study of topological models appropriate for the denotational semantics of programming languages [see ibid., 183-197 (1994; Zbl 0911.54025)].

It follows from the authors’ results that each \(T_0\)-space with a \(\sigma\)-disjoint base admits a weightable quasi-metric and that each weightable quasi-metric space is quasi-developable. Those partially ordered sets whose Alexandrov topology admits a weightable quasi-metric are characterized. The authors also show that the Pixley-Roy space over the reals does not admit a weightable quasi-metric.

For the entire collection see [Zbl 0903.00047].

It follows from the authors’ results that each \(T_0\)-space with a \(\sigma\)-disjoint base admits a weightable quasi-metric and that each weightable quasi-metric space is quasi-developable. Those partially ordered sets whose Alexandrov topology admits a weightable quasi-metric are characterized. The authors also show that the Pixley-Roy space over the reals does not admit a weightable quasi-metric.

For the entire collection see [Zbl 0903.00047].

##### MSC:

54E15 | Uniform structures and generalizations |

54E35 | Metric spaces, metrizability |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

68Q55 | Semantics in the theory of computing |