This book contains selected chapters on recent research in topology. It bridges the gap between recent trends of topological theories and their applications in areas like social sciences, natural sciences, soft computing, economics, theoretical chemistry, cryptography, pattern recognitions and granular computing. There are 14 chapters, including two chapters on mathematical economics from the perspective of topology. The book discusses topics on function spaces, relator space, preorder, quasi-uniformities, bitopological dynamical systems, b-metric spaces and related fixed point theory. This book is useful to researchers, experts and scientists in studying the cutting-edge research in topology and related areas and helps them applying topology in solving real-life problems the society and science are facing these days.
Author(s): Santanu Acharjee
Series: Industrial and Applied Mathematics
Publisher: Springer-ISIAM
Year: 2023
Language: English
Pages: 263
City: Aligarh
Preface
Contents
About the Editor
1 Spaces of Minimal Usco and Minimal Cusco Maps as Fréchet Topological Vector Spaces
1.1 Introduction
1.2 Minimal Usco and Minimal Cusco Maps
1.3 When Is the Space of Minimal USCO/CUSCO Maps a Completely Metrizable Space?
1.4 Minimal USCO/CUSCO Maps with a Structure of a Vector Space
1.5 Isomorphism of the Spaces of Minimal Usco and Minimal Cusco Maps
1.6 Conclusions
References
2 Contra Continuity Properties of Relations in Relator Spaces
2.1 Introduction
2.2 A Few Basic Definitions on Relations
2.3 A Few Basic Definitions on Relators
2.4 A Few Basic Theorems on Relations and Relators
2.5 Some Further Theorems on Relations and Relators
2.6 Some Basic Structures Derived from Relators
2.7 Some Further Structures Derived from Relators
2.8 Some Important Closure Operations for Relators
2.9 Some Further Important Unary Operations for Relators
2.10 Proximal Interior and Closure Reversing Relations
2.11 Topological Interior and Closure Reversing Relations
2.12 Fatness and Denseness Reversing Relations
2.13 Proximal Openness and Closedness Reversing Relations
2.14 Topological Openness and Closedness Reversing Relations
2.15 Contra Continuity Properties of the Identity Function
2.16 Some Further Results on The Identity Function
2.17 Two Illustrating Examples and a Constancy Theorem
References
3 The Continuous Representation Property in Utility Theory
3.1 Introduction
3.2 Preliminaries
3.3 The Continuous Representation Property: A Review of the Main Contributions
3.4 CRP and CCC
3.5 Further Generalizations of CRP
3.6 Concluding Remarks
References
4 On Quasi-uniformities, Function Spaces and Atoms: Remarks and Some Questions
4.1 Introduction
4.2 Quasi-Uniformities on Function Spaces
4.3 Quasi-Uniformities on Function Spaces Generated by Atoms
4.4 Open Problems
References
5 Some Cardinal Estimations via the Inclusion-Exclusion Principle in Finite T0 Topological Spaces
5.1 Introduction
5.2 Preliminaries
5.3 Main Results
5.4 Conclusion
References
6 Representations of Preference Relations with Preutility Functions on Metric Spaces
6.1 Introduction
6.2 Notations, Definitions
6.3 Some Observations
6.3.1 Debreu's Theorems
6.3.2 Lexicographic Order
6.4 Flow Functions
6.5 Representation Theorems with Preutility Functions
6.6 Representation Theorems with Utility Functions
6.7 Microeconomics
6.8 Conclusion
References
7 Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems
7.1 Introduction
7.2 Preliminary Definitions
7.3 On Entropy of a Pairwise Continuous Map in NWPC Bitopological Dynamical Systems
7.4 Fundamental Properties of Entropy in NWPC Bitopological Dynamical Systems
7.5 Possible Connection to Neural Activity of Human Brain
7.6 Open Questions
7.7 Conclusion
References
8 Topological Approaches for Vector Variational Inequality Problems
8.1 Introduction
8.2 Variational Inequalities and Their Generalizations
8.3 Preliminaries
8.4 On Solutions of Vector Variational Inequality Problems
References
9 Ideals and Grills Associated with a Rough Set
9.1 Introduction
9.2 Ideal Approximation Spaces
9.3 Grill Approximation Spaces
9.4 Conclusion
References
10 Filter Versus Ideal on Topological Spaces
10.1 Introduction
10.2 Big Sets
10.3 Small Sets
10.4 Homeomorphism
10.5 Topological Cryptography and Applications of Big Set
10.6 Conclusion
References
11 Fisher Type Set-valued Mappings in b-metric Spaces and an Application to Integral Inclusion
11.1 Introduction
11.2 Main Results
11.3 Stability of Fixed Point Sets
11.4 Application to Fredholm Type Integral Inclusion
References
12 Topological Aspects of Granular Computing
12.1 Introduction
12.2 Granular Computing in Binary Relations
12.2.1 Mathematical Structure of Binary Granulation
12.2.2 Neighborhood Systems and Granular Computing Models
12.3 Algebraic Quotient Space in Granular Computing
12.3.1 Algebraic Quotient Space Model
12.3.2 Algebraic Operator-Based Quotient Map
12.4 Interactive Granular Computing and Neighborhood Systems
12.5 Some Open Questions Inspired by Microscopy, Biology and Neuroscience
12.6 Conclusion
References
13 On Topological Index of Naturally Occurring Zeolite Material [4, n]
13.1 Introduction
13.2 Preliminaries
13.3 Main Results
13.4 Neighbourhood-Based Topological Index
13.5 Conclusion
References
14 q-Rung Orthopair Fuzzy Points and Applications to q-Rung Orthopair Fuzzy Topological Spaces and Pattern Recognition
14.1 Introduction
14.2 q-Rung Orthopair Fuzzy Points
14.3 A Pattern Recognition Application
14.3.1 A Dice Similarity Measure and A Distance Measure
14.3.2 Pattern Recognition
14.4 Continuity and Convergence
14.5 Conclusion
References
