Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications

1st Edition, Volume 199 - January 20, 2005
Author: Badri Dvalishvili
Language: English
Hardback ISBN:
9 7 8 - 0 - 4 4 4 - 5 1 7 9 3 - 7
eBook ISBN:
9 7 8 - 0 - 0 8 - 0 4 5 9 4 6 - 2

This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological… Read more

Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications

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This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, and theory of ordered topological spaces. Moreover, a high level of modern knowledge of bitopological spaces theory has made it possible to introduce and study algebra of new type, the corresponding representation of which brings one to the special class of bitopological spaces. It is beyond any doubt that in the nearest future the areas of essential applications will be the theories of linear topological spaces and topological groups, algebraic and differential topologies, the homotopy theory, not to mention other fundamental areas of modern mathematics such as geometry, mathematical logic, the probability theory and many other areas, including those of applied nature. Key Features:- First monograph is "Generalized Lattices"

North-Holland Mathematical Studies 199

Dedication

Preface

Acknowledgements

Chapter 0: Preliminaries

0.1 Symbols and Notations. Basic Concepts of Bitopology

0.2 Internal Characterization of Pairwise Complete Regularity

Chapter I: Different Families of Sets in Bitopological Spaces

1.1 (i, j)-Nowhere Dense Sets and (i, j)-Category Notions

1.2 (i, j)-Locally Closed Sets

1.3 (i,j)-Boundaries. (i,j)-Open Domains and (i,j)-Closed Domains. (i, j)-Semiopen Sets and (i,j)-Semiclosed Sets, (i,j)-Semiopen Domains and (i, j)-Semiclosed Domains

1.4 Sets Pairwise Dense in Themselves. (i, j)-Perfect Sets and Pairwise Scattered Sets

1.5 Relative Properties

Chapter II: Different Relations Between Two Topologies on a Set and Bitopological Insertions

2.1 The S-Relation

2.2 The C-Relation

2.3 The N-Relation

2.4 Bitopological Insertions

Chapter III: Dimension of Bitopological Spaces

3.1 Pairwise Small Inductive Dimension

3.2 Pairwise Large Inductive Dimension

3.3 Pairwise Covering Dimension

Chapter IV: Baire-Like Properties of Bitopological Spaces

4.1 (1, 2)-Baire Spaces

4.2 Almost-(2,1)-Baire Spaces

4.3 Strong Baire-Like Properties

4.4 Some Modifications of Baire-Like Properties

Chapter V: Dynamics of Bitopological Relations, Baire-Like Properties and Dimensions

5.1 Mappings of Bitopological Spaces

5.2 Mappings Applied to Some Bitopological Notions

Chapter VI: Generalized Boolean Algebras and Related Problems. Representation Theorems

6.1 Gosets, Generalized Lattices, Generalized Boolean Algebras, and the Corresponding Operations

6.2 Generalized Ideals and their Variety. Stone Family of Prime Generalized Ideals

6.3 Generalized Filters and their Variety. Generalized Duality between Generalized Ideals and Generalized Filters

6.4 Generalized Homomorphisms and Generalized Isomorphisms

6.5 (i, j)-Atoms and Pairwise Atomic Generalized Boolean Algebras

6.6 Generalized Boolean Factor Algebras

6.7 Generalized Fields of Sets and the Generalized Field Representation of a Generalized Boolean Algebra

6.8 Bitopological Representation of a Generalized Boolean Algebra

6.9 Generalized Boolean Rings

Chapter VII: Applications of Bitopologies

7.1 Applications in Analysis

Relations with Potential Theory

7.3 Applications in General Topology and Theory of Ordered Topological Spaces

Bibliography

List of Special Symbols and Notations

Topologies

Spaces

Axioms of Separation

Dimension Functions

Families of Sets

Classes of Functions

Index

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Badri Dvalishvili