Relationships between completeness of fuzzy quasi-uniform spaces

Document Type: Original Manuscript

Authors

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

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

Abstract

In this paper, we give a kind of Cauchy 1-completeness in probabilistic quasi-uniform spaces by using 1-filters. Utilizing
the relationships among probabilistic quasi-uniformities, classical quasi-uniformities and Hutton [0
, 1]-quasi-uniformities,
we show the relationships between their completeness. In fuzzy quasi-metric spaces, we establish the relationships
between the completeness of induced probabilistic quasi-uniform spaces and both completeness of induced classical
quasi-uniform spaces and induced Hutton [0
, 1]-quasi-uniform spaces.

Keywords


[1] J. Guti´errez Garc´ıa, M. A. De Prada Vicente, Hutton [0, 1]-quasi-uniformities induced by fuzzy (quasi-)metric spaces, Fuzzy Sets and Systems, 157 (2006), 755-766.
[2] J. Guti´errez Garc´ıa,
On stratified L-valued filters induced by -filters, Fuzzy Sets and Systems, 157 (2006), 813-819.
[3] J. Guti´errez Garc´ıa, M. A. De Prada Vicente,
Characteristic values of -filter, Fuzzy Sets and Systems, 156 (2005),
55-67.
[4] J. Guti´errez Garc´ıa,
A unified approach to the concept of fuzzy L-uniform space, Ph.D Thesis, 2000.
[5] J. Guti´errez Garc´ıa, M. A. De Prada Vicente, S. Romaguera,
Completeness of Hutton [0,1]-quasi-uniform spaces,
Fuzzy Sets and Systems,
158 (2007), 1791-1802.
[6] J. Guti´errez Garc´ıa, J. R. -Lopez, S. Romaguera,
On fuzzy uniformities induced by a fuzzy metric space, Fuzzy Sets
and Systems,
330 (2018), 52-78.
[7] A. George, P. Veeramani,
On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399.
[8] A. George, P. Veeramani,
Some theorems in fuzzy metric spaces, Journal of Fuzzy Mathematica, 3 (1995), 933-940.
[9] V. Gregori, S. Romaguera,
On completion of fuzzy metric spaces, Fuzzy Sets and Systems, 130 (2002), 399-404.
[10] V. Gregori, A. Lopez-Crevillen, S. Morillas, A. Sapena,
On convergence in fuzzy metric spaces, Topology and its
Applications,
156 (2009), 3002-3006.
[11] V. Gregori, S. Romaguera,
Fuzzy quasi-metric spaces, Applied General Topology, 5 (2004), 129-136.
[12] B. Hutton,
Uniformities on fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 58 (1977),
559-571.
[13] U. H¨ohle, A. Sostak, ˘
Axiomatic foundations of fixed-basis fuzzy topology, in: U. H¨ohle, S. E. Rodabaugh (Eds.),
Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3,
Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 123-272 (Chapter 3).
[14] U. H¨ohle,
Probabilistic metrization of fuzzy uniformities, Fuzzy Sets and Systems, 8(1) (1982), 63-69.
[15] U. H¨ohle,
Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Mathematica , 38 (1982), 289-323.
[16] I. Kramosil, J. Michalek,
Fuzzy metrics and statistical metric spaces, Kybernetika, 11(5) (1975), 336-344.
[17] R. Lowen,
Fuzzy uniform spaces, Journal of Mathematical Analysis and Applications, 82 (1981), 370-385.
[18] R. Lowen,
Completeness, compactness and precompactness in fuzzy uniform spaces: part I, Journal of Mathematical
Analysis and Applications,
90 (1982), 563-581.
[19] B. Pang,
Degrees of separation properties in stratified L-generalized convergence spaces using residual implication,
Filomat,
31(20) (2017), 6293-6305.
[20] B. Pang,
Stratified L-ordered filter spaces, Quaestiones Mathematicae, 40(5) (2017), 661-678.
[21] B. Pang, F. G. Shi,
Subcategories of the category of L-convex spaces, Fuzzy Sets and Systems, 313 (2017), 61-74.
[22] B. Pang, F. G. Shi,
Strong inclusion orders between L-subsets and its applications in L-convex spaces, Quaestiones
Mathematicae,
41(8) (2018), 1021-1043.
[23] J. L. Sieber, W. J. Pervin,
Completeness in quasi-uniform spaces, Mathematische Annalen, 158 (1965), 79-81.
[24] B. Schweizer, A. Sklar,
Statistical metric spaces, Pacific Journal of Mathematics, 10 (1960), 314-334.
[25] F. G. Shi, Pointwise quasi-uniformities and p. q. metrics on completely distributive lattices, Acta Mathematica
Sinica,
39 (1996), 701-706.
[26] Z. Y. Xiu, B. Pang,
M-fuzzifying cotopological spaces and M-fuzzifying convex spaces as M-fuzzifying closure
spaces
, Journal of Intelligent & Fuzzy Systems, 33 (2017), 613-620.
[27] Z. Y. Xiu, B. Pang,
Base axioms and subbase axioms in M-fuzzifying convex spaces, Journal of Intelligent & Fuzzy
Systems,
15(2) (2018), 75-87.
[28] Y. Yue, F. G. Shi,
On fuzzy pseudo-metric spaces, Fuzzy Sets and Systems, 161 (2010), 1105-1106.
[29] Y. Yue, J. Fang,
Uniformities in fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 12 (2015), 43-57.
[30] Y. Yue, J. Fang,
Completeness in probabilistic quasi-uniform space, Fuzzy Sets and Systems, Available online 7
August 2018.
[31] W. Yao, L. X. Lu,
Analogizing Hutton’s quasi-uniformities for complete lattices and extending Shi’s quasiuniformities to closed set lattices, Fuzzy Sets and Systems, 160 (2009), 1233-1244.
[32] D. Zhang,
A comparison of various uniformities in fuzzy topology, Fuzzy Sets and Systems, 140 (2003), 399-422.
[33] D. Zhang,
Uniform environments as a general framework for metrics and uniformities, Fuzzy Sets and Systems,
159 (2008), 559-572.