Domain theoretic characterisations of quasi-metric completeness in terms of formal balls†

Authors:
Salvador Romaguera;Oscar Valero
Affiliations:
Instituto universitario de matemática pura y aplicada, universidad politécnica de valencia, 46071 valencia, spain email: sromague@mat.upv.es;Departamento de ciencias matemáticas e informática, universidad de las islas baleares, 07122 palma de mallorca, baleares, spain email: o.valero@uib.es
Venue:
Mathematical Structures in Computer Science
Year:
2010

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Abstract

We characterise those quasi-metric spaces (X, d) whose poset BX of formal balls satisfies the condition (*)\begin{linenomath}\begin{equation} \text{for every } (x,r),(y,s)\in \mathbf{B}X,\ (x,r)\ll (y,s)\Leftrightarrow d(x,y) From this characterisation, we then deduce that a quasi-metric space (X, d) is Smyth-complete if and only if BX is a dcpo satisfying condition (*). We also give characterisations in terms of formal balls for sequentially Yoneda complete quasi-metric spaces and for Yoneda complete T1 quasi-metric spaces. Finally, we discuss several properties of the Heckmann quasi-metric on the formal balls of any quasi-metric space.