Partial quasi-metrics
A partial quasi-metric space \((X, p)\) consists of a set \(X\) and a distance function \(p: X\times X\to [0,\infty)\) satisfying (i) \(p(x, x)\leq\min\{p(x, y),p(y, x)\}\) whenever \(x,y\in X\), (ii) \(x= y\) iff \(p(x, x)= p(x, y)\) and \(p(y, y)= p(y, x)\) whenever \(x,y\in X\), (iii) \(p(x, z)+ p(y, y)\leq p(x, y)+ p(y, z)\) whenever \(x,y,z\in X\). Obviously, partial quasi-metrics are a simultaneous generalization of quasimetrics and of partial metrics in the sense of \textit{S. G. Matthews} [Ann. N.Y. Acad. Sci. 728, 183--197 (1994; Zbl 0911.54025)]. A triple \((X, q,w)\) is said to be a quasi-metric space with compatible weight if \((X, q)\) is a quasi-metric space and \(w: X\to [0,\infty)\) is a function satisfying \(w(y)\leq q(x, y)+ w(x)\) whenever \(x,y\in X\). It is shown that partial quasi-metric spaces and quasi-metric spaces with compatible weight are equivalent in the following sense. If \((X, p)\) is a partial quasi-metric space, then \((X, q_p,w_p)\) is a quasi-metric space with compatible weight, where \(q_p(x,y)= p(x,y)- p(x,x)\) whenever \(x,y\in X\), and \(w_p(x)= p(x,x)\) whenever \(x\in X\). Conversely, if \((X,q,w)\) is a quasi-metric space with compatible weight, then \((X, p_{qw})\) is a partial quasi-metric space, where \(p_{qw}(x,y)= q(x,y)+ w(x)\) whenever \(x,y\in X\). Using this correspondence, a partial quasi-metric space \((X, p)\) is defined to be complete if the metric space \((X,d_{q_p})\) is complete, where \(d_{q_p}(x, y)= q_p(x,y)+ q_p(y,x)\) whenever \(x,y\in X\). By applying the correspondence to a construction of \textit{S. Romaguera}, \textit{S. Oltra} and \textit{E. A. Sánchez-Pérez} [Rend. Circ. Mat. Palermo, II. Ser. 51, No. 1, 151--162 (2002; Zbl 1098.54027)] it is shown that every partial quasi-metric space has a completion. Moreover, a fixed point theorem of Matthews is extended from partial metric spaces to partial quasi-metric spaces.
scientific article; zbMATH DE number 5079014