An identification theorem for the completion of the Hausdorff fuzzy metric

Abstract We prove that given a fuzzy metric space (in the sense of Kramosil and Michalek), the completion of its Hausdorff fuzzy metric space is isometric to the Hausdorff fuzzy metric space of its completion, when the Hausdorff fuzzy metrics are defined on the respective collections of non-empty closed subsets. As a consequence, we deduce the corresponding result for metric spaces by using the standard fuzzy metric. An application to the extension of multivalued mappings and some illustrative examples are also given.

[1]  Francisco Castro-Company,et al.  The Banach Contraction Principle in Fuzzy Quasi-metric Spaces and in Product Complexity Spaces: Two Approaches to Study the Cost of Algorithms with a Finite System of Recurrence Equations , 2010, IJCCI.

[2]  Ivan Kramosil,et al.  Fuzzy metrics and statistical metric spaces , 1975, Kybernetika.

[3]  Valentín Gregory,et al.  Some properites of fuzzy metric spaces , 2000 .

[4]  Manuel Sanchis,et al.  Ultrafilter-limit points in metric dynamical systems , 2007 .

[5]  Radko Mesiar,et al.  A Fixed Point Theorem in Probabilistic Metric Spaces and an Application , 1996 .

[6]  Francisco Castro-Company,et al.  The bicompletion of fuzzy quasi-metric spaces , 2011, Fuzzy Sets Syst..

[7]  H. Sherwood,et al.  On the completion of probabilistic metric spaces , 1966 .

[8]  Andreas Blass.  Ultrafilters: Where topological dynamics = algebra = combinatorics , 1993 .

[9]  Gerald Beer,et al.  Topologies on Closed and Closed Convex Sets , 1993 .

[10]  Russell J. Egbert,et al.  Products and quotients of probabilistic metric spaces , 1968 .

[11]  Robert M. Tardiff,et al.  Topologies for probabilistic metric spaces , 1976 .

[12]  P. Veeramani,et al.  On some results of analysis for fuzzy metric spaces , 1997, Fuzzy Sets Syst..

[13]  Valentín Gregori,et al.  On completion of fuzzy metric spaces , 2002, Fuzzy Sets Syst..

[14]  Valentín Gregori,et al.  Some properties of fuzzy metric spaces , 2000, Fuzzy Sets Syst..

[15]  Endre Pap,et al.  Fixed Point Theory in Probabilistic Metric Spaces , 2001 .

[16]  Salvador Romaguera,et al.  The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words , 2007 .

[17]  A. George,et al.  On some results in fuzzy metric spaces , 1994 .

[18]  Ethan Akin.  Recurrence in topological dynamics , 1997 .

[19]  Norman R. Howes,et al.  Modern Analysis and Topology , 1995 .

[20]  On Nonsymmetric Topological and Probabilistic Structures , 2009 .

[21]  Y. Cho,et al.  Quicksort algorithm: Application of a fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words , 2009 .

[22]  B. Schweizer,et al.  Statistical metric spaces. , 1960 .

[23]  J. W. deBakker,et al.  Control flow semantics , 1996 .

[24]  Salvador Romaguera,et al.  The Hausdorff fuzzy quasi-metric , 2010, Fuzzy Sets Syst..

[25]  S. Nadler.  Multi-valued contraction mappings. , 1969 .

[26]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[27]  Endre Pap,et al.  A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces , 2002, Fuzzy Sets Syst..

[28]  Ethan Akin.  Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions , 2013 .

[29]  E. Michael.  Topologies on spaces of subsets , 1951 .

[30]  Viorel Radu,et al.  Some fixed point theorems probabilistic metric spaces , 1987 .

[31]  Salvador Romaguera,et al.  Contraction Maps on Ifqm-spaces with Application to Recurrence Equations of Quicksort , 2006, MFCSIT.

[32]  V. H. Badshah,et al.  On Some Results in Fuzzy Metric Spaces , 2014 .