The quasimetrization problem in the (bi)topological spaces
A quasi-pseudometric is a pair \((X,d)\), where \(X\) is a nonempty set and \(d\) is a mapping from \(X\times X\) into the real numbers \(\mathbb R\) (called a quasi-metric) satisfying for all \(x,y,z\in X\): (i) \(d(x,y)\geq 0\), (ii) \(d(x,x)= 0\), (iii) \(d(x,z)\leq (x,z)+ d(z,y)\). If \(d(x,y)= 0\) if and only if \(x= y\), then \(d\) is called a quasi-metric on \(X\). The sets \(B(x,r)= \{z\mid d(x,y)< r\}\) constitute a base for a topology \(\tau_d\). If \(d\) is a quasi-pseudometric on \(X\), \(d^{-1}(x,y)= d(x,y)\) is also a quasi-pseudometric on \(X\). Thus a quasi-pseudometric determines two topologies \(\tau_d\) and \(\tau_{d-1}\). The quasi-(pseudo)metrization problem for a bitopological space \((X,\tau_0,\tau_1)\) is to find necessary and sufficient conditions for \(\tau_0= \tau_d\) and \(\tau_1= \tau_{d-1}\) for some quasi-(pseudo)metric. Related to the problem of quasi-pseudometrizability of a bitopological space is that of quasi-pseudometrizability of a topological space. In this paper, in the first section, the author gives in Theorem 1.6 the solution of the inverse problem raised by \textit{J. Williams} [Trans. Am. Math. Soc. 168, 435--469 (1972; Zbl 0221.54022)]. In the second section a theorem on the necessary and sufficient conditions for a bitopological space to be quasi-metrizable (Theorem 2.9) is proved, as well as an alternative form of that theorem. The author also obtains, as immediate corollaries, all the related known results. In the final section, the author gives necessary and sufficient topological conditions for a topological space to admit a quasi-pseudometric [Problem C, \textit{P. Fletcher} and \textit{W. F. Lindgren}, Quasi-uniform spaces. Lecture Notes in Pure and Applied Mathematics, 77. New York-Basel: Marcel Dekker (1982; Zbl 0501.54018)].
scientific article; zbMATH DE number 5282483
scientific article