Uniformities in fuzzy metric spaces

Document Type : Research Paper

Authors

Department of Mathematics, Ocean University of China, 238 Songling Road, 266100, Qingdao, P.R.China

Abstract

The aim of this paper is to study induced (quasi-)uniformities in Kramosil and Michalek's fuzzy metric spaces. Firstly, $I$-uniformity in the sense of J. Guti'{e}rrez  Garc'{i}a and $I$-neighborhood system in the sense of H"{o}hle and u{S}ostak are induced by the given fuzzy metric. It is shown that the fuzzy metric and the induced $I$-uniformity will generate the same $I$-neighborhood system. Secondly, the relationship between Hutton quasi-uniformities and $I$-quasi-uniformities is given and it is proved that the category of strongly stratified $I$-quasi-uniform spaces can be embedded in the category of Hutton quasi-uniform spaces as a bicoreflective subcategory. Also it is shown that two kinds of Hutton quasi-uniformities can generate the same $I$-uniformity in fuzzy metric spaces.

Keywords


[1] J. Fang, Relationships between L-ordered convergence structures and strong L-topologies,
Fuzzy Sets and Systems, 161 (2010), 2923-2944.
[2] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,
64 (1994), 395-399.
[3] A. George and P. Veeramani, Some theorems in fuzzy metric spaces, J. Fuzzy Math., 3 (1995),
933-940.
[4] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets
and Systems, 90 (1997), 365-368.
[5] V. Gregori and S. Romaguera, On completion of fuzzy metric spaces, Fuzzy Sets and Systems,
130 (2002), 399-404.
[6] V. Gregori, A. Lopez-Crevillen, S. Morillas and A. Sapena, On convergence in fuzzy metric
spaces, Topology and its Applications, 156 (2009), 3002-3006.
[7] J. Gutierrez Garca, M. A. de Prada Vicente and A. P. Sostak, A uni ed approach to the
concept of fuzzy L-unifom space, Chapter 3 in S. E. Rodabaugh, E. P. Klement, eds., Topological
and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Development in
the Mathematics of Fuzzy Sets, Trends in Logic 20(2003), Kluwer Academic Publishers
(Boston/Dordrecht/London).
[8] J. Gutierrez Garca, A uni ed approach to the concept a fuzzy L-unifom space, Ph.D Thesis,
2000.
[9] J. Gutierrez Garca and M. A. de Prada Vicente, Hutton [0, 1]-quasi-uniformities induced by
fuzzy (quasi-)metric spaces, Fuzzy Sets and Systems, 157 (2006), 755-766.
[10] U. Hohle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math., 38
(1982), 289-323.
[11] U. Hohle, Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems, 8(1)
(1982,) 63-69.
[12] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, pp.123-272,
Chapter 3 in U. Hohle and S. E. Rodabaugh, eds, Mathematics of Fuzzy Sets: Logic, Topology,
and Measure Theory, The Handbooks of Fuzzy Sets Series, Volume 3 (1999), Kluwer Academic
Publishers (Boston/Dordrecht/London).
[13] B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58 (1977), 559-571.
[14] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11(5)
(1975), 336-344.
[15] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A., 28 (1942), 535-537.
[16] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland, NewYork, 1983.
[17] Y. Yue and F. G. Shi, On fuzzy pseudo-metric spaces, Fuzzy Sets and Systems, 161 (2010),
1105-1106.
[18] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems,
158 (2007), 349{366.