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Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. (English) Zbl 1193.54016
The authors characterize those quasi-metric spaces $(X,d)$ whose poset $BX$ of formal balls satisfies the condition that for every $(x,r),(y,s)\in BX,$ $$(x,r)\ll (y,s)\Leftrightarrow d(x,y)<r-s.\tag $*$ $$ They use this characterization to deduce that a quasi-metric space $(X,d)$ is Smyth-complete if and only if $BX$ is a dcpo satisfying condition $(*)$. It follows that a quasi-metric space $(X,d)$ is Smyth-complete if and only if $(BX,\sqsubseteq)$ is a domain whose Scott topology is induced by the Heckmann quasi-metric $d^H((x,r),(y,s))=\max\{d(x,y),\vert r-s\vert\}+s-r.$ The article contains numerous other observations and results, for instance: A $T_1$ quasi-metric space is Yoneda-complete if and only if it is sequentially Yoneda-complete. The poset of formal balls of the Sorgenfrey quasi-metric space provides an example of a domain that does not satisfy condition $(*).$

54E35Metric spaces, metrizability
06B35Continuous lattices and posets, applications
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