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

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