On Nonsymmetric Topological and Probabilistic StructuresNova Publishers, 2006 - 210 pages In this book, generally speaking, some properties of bitopological spaces generated by certain non-symmetric functions are studied. These functions, called "probabilistic quasi-pseudo-metrics" and "fuzzy quasi-pseudo-metrics", are generalisations of classical quasi-pseudo metrics. For the sake of completeness as well as for convenience and easy comparison, most of the introductory paragraphs are mainly devoted to fundamental notions and results from the classical -- deterministic or symmetric -- theory. |
Table des matières
Introduction | 1 |
Preliminary Notions and Results 21 tsNorms and their Properties | 5 |
22 Triangular Norms | 7 |
23 Some Spaces of Monotone Functions | 20 |
24 QuasiInverses of Nondecreasing Functions | 25 |
25 t+ Norms and their Properties | 28 |
Probabilistic QuasiPseudoMetric Spaces 31 Probabilistic Metric Spaces | 33 |
32 Properties of QuasiPseudoSerstnev Spaces | 40 |
75 PseudoMetrization in PgpMSpaces | 91 |
76 PUniformities Generated by Deterministic MetricLike Functions | 92 |
77 A Representation Theorem for QuasiMetric Spaces | 98 |
Probabilistic Metrics and Deterministic Distance Functions 81 MetricLike Functions Defined by Probabilistic Metrics | 101 |
82 A General Method of Introducing Topologies | 109 |
83 A Family of ProSymmetrics for Distribution Functions | 113 |
Completeness in PgpMSpaces and in PtfSpaces | 117 |
91 Completeness in PqpMSpaces | 118 |
33 Relations Generated by PqpMetrics | 42 |
34 Some Classes of Probabilistic qpMetrics | 43 |
35 Families of QuasiPseudoMetrics Generated by gpSMetrics | 45 |
36 Notes and References | 49 |
Examples of Spaces 41 PDiscrete and PSimple PgpMSpaces | 51 |
42 gpSSpaces of ap9Type | 54 |
43 PqpMSpaces Generated by Probability Spaces and Stationary Markov Chains | 55 |
Topologies Generated by Metrics 51 Topological Spaces | 57 |
52 Topologies on PgpMSpaces | 60 |
53 Separation Axioms in PqpMSpaces | 65 |
Random Normed Structures 61 Random SNormed Spaces | 69 |
62 Random fNormed Structures | 74 |
63 A Characterization of The tNorms of HadzicType | 75 |
64 Remarks on QuasiNormed and Random Normed Structures | 76 |
65 Random QuasiNorms | 79 |
QuasiPseudoMetrization in Spaces 71 QuasiUniformities and QuasiEcarts | 81 |
72 QuasiMetrization | 83 |
73 Properties of PHSpaces | 86 |
74 Relationships between PH and PqpMSpaces | 88 |
92 Completeness in PHSpaces | 120 |
Fuzzy QuasiPseudoMetric Spaces 101 Fuzzy SemiMetrics | 127 |
102 Fuzzy PseudoMetrics | 130 |
103 Fuzzy QuasiPseudoSerstnev Spaces | 135 |
104 Relations Generated by FgpMetrics | 139 |
105 Some Classes of Fuzzy gpMetrics | 140 |
106 Families of Quasiecarts Generated by Fuzzy gp5Metrics | 142 |
107 Fuzzy QuasiPseudoMetric Spaces and Generalized PqpMSpaces | 145 |
108 Generalized QuasiPseudoMetric Spaces | 147 |
Probabilistic and Fuzzy Contractive Mappings 111 Some Remarks on Probabilistic Contractions | 153 |
114 BContractions on Fuzzy Menger Spaces | 161 |
115 HicksType Contractions on Fuzzy Menger Spaces | 170 |
116 Nonlinear Equations for Fuzzy Mappings | 171 |
121 Cartesian Products of FgpMSpaces | 181 |
122 Cartesian Product of FgpMSpaces of Type kn | 185 |
123 Cartesian Products of PHSpaces of Type sn | 187 |
References | 189 |
Subject Index | 203 |
Expressions et termes fréquents
apply Archimedean Assume base called CHAPTER Clearly completes the proof concept conjugate consequence consider contraction convergent Corollary defined Definition denoted easy element equivalent Example exists fact fixed point following conditions formula Fry(t function fuzzy fuzzy Menger space Fxy(t give given hence holds immediately implies increasing induced introduce Lemma Let X mapping Math means Menger space Moreover non-Archimedean nondecreasing observe obtain operation P-Cauchy sequence P-convergent P₁ pair PH)-space PM-spaces Pqp-metric PqpM-Spaces probability properties prove pseudo-metric qpŠ-space quasi-metric quasi-pseudo-metric space quasi-uniform Radu random normed relation Remark resp respect satisfies the condition Schweizer sequence Šerstnev Sklar strictly structure subset suppose t-norm t₁ Theorem theory topological space topology triangle inequality uniform unique verifies
Fréquemment cités
Page 193 - HPA Künzi, M. Mrsevic, IL Reilly and MK Vamanamurthy, Convergence, precompactness and symmetry in quasi-uniform spaces, Math.
Page 193 - Computation over Fuzzy Quantities", Florida, CRC Press, 1994. [10] O. Kaleva and S. Seikkala, "On Fuzzy Metric Spaces", Fuzzy Sets and Systems, 12, 1984, 215-229.