Quasi-uniform isomorphisms in fuzzy quasi-metric spaces, bicompletion and D-completion

Abstract

V. Gregori and S. Romaguera [17] obtained an example of a fuzzy metric space (in the sense of A. George and P. Veeramani) that is not completable, i.e. it is not isometric to a dense subspace of any complete fuzzy metric space; therefore, and contrary to the classical case, there exist quiet fuzzy quasi-metric spaces that are not bicompletable neither D-completable, via (quasi-)isometries. In this paper we show that, nevertheless, it is possible to obtain solutions to the problem of completion of fuzzy quasi-metric spaces by using quasi-uniform isomorphisms instead of (quasi-)isometries. Such solutions are deduced from a general method, given here, to obtain extension properties of fuzzy quasi-metric spaces from the corresponding ones of the classical theory of quasi-uniform and quasi-metric spaces.

Formal balls in fuzzy quasi-metric spaces

“…The bicompleteness is one of the most important completeness theory for fuzzy quasi-metric spaces [2,9]. Romaguera et al solved the bicompletion problem of fuzzy quasi-metric spaces and convinced us in [22] that it is possible to construct satisfactory bicompleteness theory for fuzzy quasi-metric spaces.…”

“…Question 2 How to construct a satisfactory Yoneda completion for any fuzzy quasi-metric space in the sense of [22]?…”

“…Motivated by [6], in this paper we first adopt the left K-Cauchy net, which is "asymmetric Cauchy", to define the Yoneda completeness and Smyth completeness on fuzzy quasi-metric spaces. The relationship among these two types of completeness and other two types of completeness on fuzzy quasi-metric spaces appeared in [5,6,9,22] are also discussed. Furthermore, we study the relationship between completeness of fuzzy (quasi-)metric spaces and directed completeness of the associated posets of formal balls.…”

A note on local bases and convergence in fuzzy metric spaces

“…In [2,8] the authors show that the class of topological spaces which are fuzzy metrizable agrees with the class of metrizable spaces. Later, several authors have contributed to the development of this theory, for instance [8,10,11,[16][17][18]]. …”

On balancedness and D-completeness of the space of semi-Lipschitz functions

“…As an application of these results to asymmetric functional analysis, we deduce that the dual space of a T 1 quasi-normed linear space is balanced and Doitchinov complete. It is interesting to recall that the study of balanced quasi-metric spaces from a fuzzy point of view has been recently started in [8,19], and that, on the other hand, some applications of balanced (extended) quasi-metrics to theoretical computer science have been given in [14,15].…”