Topology Proceedings 33 (2009), pp. 41-54

A construction method for partial metrics

Dieter Spreen

We present a general construction that starts from a family of interior-preserving open coverings of a given subspace and results in a partial metric with respect to which all subspace elements have self-distance zero. A necessary and sufficient condit ion is derived for when this partial metric induces the given topology. The condition is particularly satisfied if the members of each covering are pairwise disjoint.

The method is based on Fletcher's universal construction for transitive quasi-uniformities. Important examples of partial metrics in the literature can be obtained in this way.

As a consequence of the construction, the set of all points with self-distance zero is a G_d. Moreover, this subspace is zero-dimensional in its induced topology.

Keywords: partial metric, quasi-uniformity, asymmetric topology, domain theory

Mathematics Subject Classification: 54E15 (06B35 54E25 54E35 68Q55)

Document formats: PDF file 176.4 Kb

Topology Proceedings, Volume 33 (2009)
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