The quasimetrization problem in the (bi)topological spaces
type: Item
instance of: scholarly article
title: The quasimetrization problem in the (bi)topological spaces
zbMATH Open document ID: 1147.54014
DOI: 10.1155/2007/76904
author: Athanasios Andrikopoulos
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/54610
review text: 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)].
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 5282483
zbMATH Keywords: quasi-metric
zbMATH Keywords: quasi-pseudometric
zbMATH Keywords: quasimetrization problem
zbMATH Keywords: topological space
zbMATH Keywords: bitopological space
zbMATH Keywords: pairwise regular space
MaRDI profile type: Publication
OpenAlex ID: W1971474583
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description: scientific article; zbMATH DE number 5282483