The Katětov construction modified for a \(T_0\)-quasi-metric space (Q409506)

Recall that a pair \((X, d)\) is called a \(T_0\)-quasi-metric space if \(X\) is a set and \(d: X\times X\to[0,\infty)\) is a mapping satisfyingNEWLINENEWLINE (i) \(d(x, x)= 0\) whenever \(x\in X\);NEWLINENEWLINE (ii) \(d(x, z)\leq d(x, y)+ d(y, z)\) whenever \(x,y,z\in X\); andNEWLINENEWLINE (iii) \(d(x, y)= 0= d(y, x)\) implies that \(x=y\).NEWLINENEWLINE If \((X, d)\) is a \(T_0\)-quasi-metric space, then \((X, d^S)\) is a metric space, where \(d^S= \sup\{d, d^{-1}\}\) and \(d^{-1}(x, y)= d(y, x)\) whenever \(x,y\in X\). A \(T_0\)-quasi-metric space \((X, d)\) is called bicomplete (respectively, supseparable), if \((X, d^S)\) is complete (respectively, separable). It is shown that there exists a unique (up to isometry) \(T_0\)-quasi-metric space \(q\mathbb{U}\) with the following properties:NEWLINENEWLINE (a) \(q\mathbb{U}\) is bicomplete and supseparable;NEWLINENEWLINE (b) every isometry between two finite subspaces of \(q\mathbb{U}\) extends to an isometry of \(q\mathbb{U}\) onto itself; andNEWLINENEWLINE (c) \(q\mathbb{U}\) contains an isometric copy of every supseparable \(T_0\)-quasi-metric space.NEWLINENEWLINE The construction of \(q\mathbb{U}\) is similar to \textit{M. Katĕtov's} construction of Urysohn's famous universal metric space \(\mathbb{U}\) in [General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 323--330 (1988; Zbl 0642.54021)].