We extend the celebrated result of W. A. Kirk that a metric space "Equation missing" is complete if and only if every Caristi self-mapping for "Equation missing" has a fixed point, to partial metric spaces.

Given a partial metric space (X,p), we use (BX, ⊑_dp) to denote the poset of formal balls of the associated quasi-metric space (X,d_p). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑_dp). In particular, we prove that a partial metric space (X,p) is complete if and only if the poset (BX, ⊑_dp) is a domain. Furthermore, for any complete partial metric space (X,p), we construct a Smyth complete quasi-metricqonBXthat extends the quasi-metricd_psuch that both the Scott topology and the partial order ⊑_dpare induced byq. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Künzi, H. Pajoohesh and M. P. Schellekens (Künziet al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.