A Construction Method for Partial Metrics

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 condition 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 selfdistance zero is a Gδ. Moreover, this subspace is zero-dimensional in its induced topology.